DEVELOPING AND OPTIMIZING THE REPLICA EXCHANGE-BASED FREE ENERGYMETHODS
By
DANIAL SABRI DASHTI
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2013
c⃝ 2013 Danial Sabri Dashti
2
I dedicate this dissertation to my wife and my family.
3
ACKNOWLEDGMENTS
I primarily thank my advisor, Professor Adrian Roitberg for our scientific conversations
and discussions. I am also grateful for significant assistance of my lovely wife, Sahar,
during my Ph.D. period. I would never have been able to finish my dissertation without
the support of my family. I also like to thank Dr. Yilin Meng for his effective collaboration.
Moreover, thanks go out to all who supported me over my Ph.D study at the University of
Florida.
4
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 Free Energies and Ensembles . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.1 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.2 Partition Functions and Ensembles . . . . . . . . . . . . . . . . . . 131.2.3 Free Energy and Potential of Mean Force . . . . . . . . . . . . . . 14
1.2.3.1 Helmholtz and Gibbs free energies . . . . . . . . . . . . . 141.2.3.2 Potential of mean force . . . . . . . . . . . . . . . . . . . 15
1.3 Sampling in Biomolecular . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.1 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.2 Standard Sampling Techniques in Biomolecular Simulations . . . . 16
1.3.2.1 Molecular dynamics . . . . . . . . . . . . . . . . . . . . . 161.3.2.2 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Advanced Sampling Methods . . . . . . . . . . . . . . . . . . . . . . . . . 181.4.1 Generalized Ensemble Methods . . . . . . . . . . . . . . . . . . . . 18
1.4.1.1 Simulated tempering . . . . . . . . . . . . . . . . . . . . . 181.4.1.2 Multicanonical algorithm . . . . . . . . . . . . . . . . . . . 191.4.1.3 Wang-Landau sampling . . . . . . . . . . . . . . . . . . . 191.4.1.4 Replica exchange . . . . . . . . . . . . . . . . . . . . . . 201.4.1.5 Umbrella sampling . . . . . . . . . . . . . . . . . . . . . . 22
1.4.2 Slow-Growth Methods . . . . . . . . . . . . . . . . . . . . . . . . . 221.4.2.1 Thermodynamic integration . . . . . . . . . . . . . . . . . 231.4.2.2 Free energy perturbation . . . . . . . . . . . . . . . . . . 24
1.5 Convergence, Error Estimation, and Sampling Quality in BiomolecularSimulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.5.1 Error in Computational Methods . . . . . . . . . . . . . . . . . . . . 241.5.2 Ergodicity and Convergence . . . . . . . . . . . . . . . . . . . . . . 251.5.3 Sampling Quality Checks . . . . . . . . . . . . . . . . . . . . . . . 26
1.5.3.1 Root mean square deviation analysis . . . . . . . . . . . 261.5.3.2 Root mean square deviation clustering . . . . . . . . . . 261.5.3.3 Block averaging . . . . . . . . . . . . . . . . . . . . . . . 271.5.3.4 Principal component analysis . . . . . . . . . . . . . . . . 271.5.3.5 Kullback-Leibler divergence . . . . . . . . . . . . . . . . . 28
5
1.6 Outline of My Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2 COMPUTING ALCHEMICAL FREE ENERGY DIFFERENCES WITH HAMILTONIANREPLICA EXCHANGE MOLECULAR DYNAMICS (HREMD) Simulations . . . 31
2.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 Theory and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.1 Free Energy Perturbation(FEP) . . . . . . . . . . . . . . . . . . . . 332.2.2 Thermodynamic Integration . . . . . . . . . . . . . . . . . . . . . . 352.2.3 Hamiltonian Replica Exchange Molecular Dynamics (HREMD) . . 362.2.4 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.1 Acceptance Ratio of HREMD Simulations . . . . . . . . . . . . . . 412.3.2 Aspartic Acid Model Compound Study . . . . . . . . . . . . . . . . 412.3.3 Study on Asp26 in Thioredoxin . . . . . . . . . . . . . . . . . . . . 432.3.4 pKa Prediction for Asp26 in Thioredoxin . . . . . . . . . . . . . . . 44
2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 LARGE PKa shifts FOR BURIED PROTONABLE RESIDUES. RATIONALIZINGTHE CASE OF GLUTAMATE 66 IN STAPHYLOCOCCAL NUCLEASE . . . . . 47
3.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 PH-REPLICA EXCHANGE MOLECULAR DYNAMICS IN PROTEINS USINGA DISCRETE PROTONATION METHOD . . . . . . . . . . . . . . . . . . . . . 62
4.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2 Theoretical Method and Simulation Details . . . . . . . . . . . . . . . . . 65
4.2.1 Constant pH Molecular Dynamics and pH-Replica Exchange MolecularDynamics (pH-REMD) . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.2 Titration Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2.3 pH-REMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4.1 Titratable Model Compounds . . . . . . . . . . . . . . . . . . . . . 694.4.2 ADFDA Model Compounds . . . . . . . . . . . . . . . . . . . . . . 694.4.3 Heptapeptide Derived from OMTKY3 . . . . . . . . . . . . . . . . . 71
4.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 OPTIMIZATION OF UMBRELLA SAMPLING REPLICA EXCHANGE MOLECULARDYNAMICS BY REPLICA POSITIONING . . . . . . . . . . . . . . . . . . . . . 78
5.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6
5.2.1 Umbrella Sampling Replica Exchange . . . . . . . . . . . . . . . . 815.2.2 Calculating Exchange Acceptance Ratio in Umbrella Sampling
Replica Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.2.3 Umbrella Sampling Replica Exchange Optimization . . . . . . . . . 835.2.4 Umbrella Sampling Replica Exchange Optimization Workflows . . 845.2.5 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.3.1 Potential of Mean Force Along the Butane Dihedral Angle . . . . . 885.3.2 Potential of Mean Force for NH+
4 + OH− Salt Bridge in ExplicitSolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
APPENDIX
A CALCULATING THE PDF OF ∆ξ IN USRE . . . . . . . . . . . . . . . . . . . . 95
B ESTIMATING MEANS AND VARIANCES OF THE WINDOWS USING NEARESTNEIGHBOR WEIGHTED AVERAGING AND REWEIGTHING . . . . . . . . . . 96
B.1 Reweighting Fitted Gaussian Distribution on Windows . . . . . . . . . . . 98B.2 Nearest Neighbor Weighted Averaging (NNWA) . . . . . . . . . . . . . . . 99
C BARZILAI AND BORWEIN OPTIMIZATION (BB METHOD) . . . . . . . . . . . 100
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7
LIST OF TABLES
Table page
2-1 Free energy difference (in Kcal/mol) between protonated and deprotonatedAspartic acids obtained from TI, REFEP, and FEP alchemical free energy simulations 43
4-1 pKas of the reference compounds computed by different methods . . . . . . . . 69
4-2 pKa prediction and Hill coefficient of fitted from the HH equation. . . . . . . . . 70
4-3 pKa values of the titratable residues in the heptapeptide derived from OMKTY3. 72
5-1 Position and EAR for four different settings of USRE calculations on the butanedihedral simulation in implicit water. . . . . . . . . . . . . . . . . . . . . . . . . 88
8
LIST OF FIGURES
Figure page
2-1 Diagrams displaying the HREMD exchange algorithm and free energy calculation. 37
2-2 Thermodynamic cycle used to compute the pKa shift. . . . . . . . . . . . . . . 39
2-3 Cumulative average free energy differences between protonated and deprotonatedaspartic acid in the model compound (∆G(AH→ A−)) . . . . . . . . . . . . . . 42
2-4 Predicted pKa value of Asp26 in thioredoxin as a function of time . . . . . . . . 45
3-1 one turn of an α−helix exposes the side chain of GLU66. . . . . . . . . . . . . 50
3-2 Free energy convergence in REFEP for unrestrained V66E ∆+PHS (red) andGlu model compound (green). . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3-3 Thermodynamic cycle for conformation protonation model of GLU66. . . . . . . 52
3-4 Cumulative Free energy VS Simulation time for Glu-66 restrained inside (Red)and restrained outside (Green). . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3-5 Concentration of species vs. pH for two different value of Ke/b,D i.e., Ke/b,D;1 =2 and Ke/b,D;2 = 108. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3-6 CSS, as defined in Equation 3–4, as a function of pH for Ke/b,D=10 . . . . . . . 56
3-7 Apparent pKa vs. Ke/b,D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3-8 Carbon α RMSD distributions of residues 62 to 69 and the side chain of GLU66in unrestrained REFEP simulations . . . . . . . . . . . . . . . . . . . . . . . . . 58
3-9 Average secondary structure for residue 57-69 vs. reaction coordinate. . . . . 59
3-10 Ellipticity at 222 nm vs. reaction coordinate . . . . . . . . . . . . . . . . . . . . 60
4-1 Comparison of Hill plots between pH-REMD and CpH methods for Lys referencemodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4-2 The titration curves of Asp side chains in ADFDA computed by both constantpH MD (blue and purple) and pH-REMD (red and green) methods. . . . . . . . 71
4-3 Cumulative average protonation fractions of Asps side chains in ADFDA VSMC titration steps at pH=4.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4-4 The titration curves of Asp3 in the heptapeptide derived from OMTKY3. . . . . 73
4-5 The titration curves of Lys5 and Tyr7 in the heptapeptide derived from OMTKY3. 74
4-6 Cumulative average protonation fraction for TYR7 versus MC titration steps atpH=8.0,9.0,10.0,11.0,12.0 and 13.0. . . . . . . . . . . . . . . . . . . . . . . . . 75
9
4-7 The Kullback-Leibler divergence measure of RMSD distributions of CpH (Green)and pH-REMD (Red) respect to the final RMSD distribution in CpH. . . . . . . 76
4-8 RMSD Autocorrelation for CpH (Green) and pH-REMD (Red) at all pHs. . . . . 77
5-1 FBSF workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5-2 SBSF workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5-3 ART probability density. The inset shows the PMF vs. reaction coordinate (dihedralangle) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5-4 Window positions vs. EARs in the equi-distance (non-optimized) (red circles)and optimized (set 2) (green circles) simulations. . . . . . . . . . . . . . . . . . 90
5-5 The mean and RMSE of Kullback-Leibler divergence over 10 simulations foreach set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
10
Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
DEVELOPING AND OPTIMIZING THE REPLICA EXCHANGE-BASED FREE ENERGYMETHODS
By
Danial Sabri Dashti
August 2013
Chair: Adrian RoitbergMajor: Physics
In recent decades, by increasing the power of computers’ hardware and developing
efficient algorithms, the study of large biomolecular systems has been facilitated. During
my PhD period, I attempted to improve the efficiency of sampling by developing new
algorithms and optimizing existing ones. My first project was about developing the
Replica Exchange Free Energy Perturbation (REFEP) method, which is a combination
of the Free Energy Perturbation (FEP) and the Hamiltonian Replica Exchange Molecular
Dynamics (HREMD) methods. We showed that the HREMD method not only improves
convergence in free energy calculations, but also can be used to estimate free energy
differences directly via the FEP algorithm. My next project was a demonstration of
the capabilities of REFEP in estimating the pKa of complicated proteins. According to
experimental measurements, the pKa value of Glutamate 66 (GLU66) in a hyperstable
mutant of staphylococcal nuclease displays a large shift, roughly 4.6 pH units, relative
to its normal value in water. In my third research project I developed and validated a
pH-Replica Exchange Molecular Dynamics (pH-REMD) method, which improves the
coupling between conformational and protonation sampling. Finally, in my last project,
I have focused on optimizing HREMD methods. The goal is to find the best position for
replicas in order to maximize the round trip between extremum positions on a replica
ladder.
11
CHAPTER 1INTRODUCTION
1.1 Prologue
Molecular simulation is a branch of statistical physics. Since biophysical systems
are too complex and inhomogeneous to be treated theoretically, the role of numerical
simulations has been recognized. With recent advancements in computer hardware
and software, the simulations of more complex molecular systems become more
feasible. There are two basic problems in studying molecular systems using computer
simulations:
• Because of the large number of particles in those systems, the size of the phasespace is enormous.
• The accuracy of the molecular models is limited.
Many methods have been proposed to address those problems. This dissertation is
about the first problem, i.e. improving the efficiency of phase space sampling. More
specifically, I concentrate on the Replica Exchange sampling method[1–4], which is one
of the most frequently used techniques for efficient sampling of the phase space.
This chapter is dedicated to the fundamentals of statistical molecular mechanics,
sampling methods, and error analysis. I start with the concept of phase space and
partition functions and finish with introducing some of the frequently used sampling
analysis techniques. Also, I provide an overview on my research projects during my
Ph.D period, and in the next chapters, describe these projects in detail.
1.2 Free Energies and Ensembles
In this section I overview the key concepts in studying of a molecular system from
classical-statistical physics’ point of view.[5–9]
1.2.1 Phase Space
According to classical statistical mechanics, the state of a system can fully be
described by knowing the positions and momenta of its particles. There are six
coordinates associated with the configuration and the momentum of every particle
12
in 3-dimensional space. Consequently, a system with N particles can be fully described
in a 6N-dimensional space, which is called phase space.
1.2.2 Partition Functions and Ensembles
An ensemble is a collection of all possible configurations of a system in phase
space. Each ensemble is concomitant with a partition function, which describes all
possible microstates of a system in equilibrium. All collective properties of the system
(e.g., average energy, entropy, and free energy) can be computed using the partition
function and its derivatives. Different types of ensembles can be defined based on the
types of communication between the system and the rest of universe. In the context of
statistical mechanics, the following are the most frequently used ensembles:
• Microcanonical ensemble (N,V, E): In a Microcanonical ensemble, volume(V), number of particles(N), and total energy (E) of a system are fixed, and thesystem has no energy communication with the universe.it can be represented asΩ(N,V, E) =
∫p
∫q
δ(H(q, p)− E) dp dq. Here, H, q, and p are the Hamiltonian, the
configuration, and the momentum of the system, respectively.
• Canonical ensemble (N,V,T): In this ensemble, V,N, and temperature (T) areconstant and the system is in thermal equilibrium with its environment. Thecanonical partition function is Q(N,V,T) =
∫p
∫q
exp(−βH(q, p))dp dq , where β =
1/kBT and kB is the Boltzmann constant. The majority of molecular simulationsare performed in this ensemble. In the case of conservative Hamiltonian, onecan separate the contributions of kinetic and potential energies. The kineticenergy contribution cancels out in most of the relative free energy calculations.Consequently it is appropriate to use configuration integral (or sum) instead ofthe full partition function, i.e., Z(N,V,T) =
∫q
exp(−βU(q)) instead of Q(N,V,T),
where U is the potential energy of the system. From now on, I will use the partitionfunction and the configuration integral interchangeably in this manuscript.
• Isothermal-isobaric ensemble (N,P,T): In this ensemble pressure (P), N ,and T are constant. The partition function can be written as Φ(N,P,T) =∫∞0
dVQ(N,V,T) exp(−βPV), which is the Laplace transform of the canonicalpartition function.
• Grand canonical ensemble(µ, V, T): In the Grand canonical ensemble, thechemical potential (µ), V, and T are fixed. The grand canonical partition functioncan be written as Ξ =
∑∞N=0 Z(N,V,T) exp(−βµN).
13
The appropriate ensemble to use depends upon the physical circumstances, although
in the limit of large systems (i.e., thermodynamic limit), where the fluctuations are
insignificant, all descriptions of ensembles become indistinguishable. The fixed values
in each ensemble are called state variables. In the next section I describe the relation
between free energies, state variables, and partition functions.
1.2.3 Free Energy and Potential of Mean Force
Free energy is the maximum amount of work that a thermodynamic system can
perform. Free energy is a state function, i.e., in each ensemble it only depends upon
the state variables of that ensemble. From the statistical mechanics point of view, in
all ensembles (except the Microcanonical ensemble), free energies are related to the
logarithm of the partition functions. Here I explain Gibbs and Helmholtz free energies,
which are frequently used in biomolecular simulations.
1.2.3.1 Helmholtz and Gibbs free energies
Helmholtz free energy is the free energy associated with the canonical ensemble
and is equal to maximum extractable work from a closed thermodynamic system at
constant volume and temperature. It is related to the canonical partition function by:
A(N,V,T) =−1β
ln(Q(N,V,T)). (1–1)
On the other hand, the Gibbs free energy (also known as free enthalpy) measures
the useful work in the isothermal-isobaric ensemble and given by:
G(N,V,T) =−1β
ln(Φ(N, P,T)). (1–2)
Since the free energy is a state function, it takes a definite, non-fluctuating value
at equilibrium. It does not give us any information about the energy of the system as a
function of some specific reaction coordinates. In biomolecular simulations, instead of
free energy, it is conventional to use potential of mean force, which is a projection of free
energy along some chemical/alchemical/physical coordinates.
14
1.2.3.2 Potential of mean force
The Potential of Mean Force (PMF) is a decomposition of the free energy along
some reaction coordinate parameters which measures how the energy of a system
changes along those coordinates. In order to calculate the PMF along some coordinates
in an ensemble, one needs to integrate the projection of the partition function over
all other degrees of freedom in configuration space. For example in the case of the
Canonical ensemble, the PMF along X can be calculated as:
PMF(X) =−1β
ln
(∫(exp(−βU(rN)δ(X− rN) drN
), (1–3)
where U(rN) is the potential energy.
1.3 Sampling in Biomolecular
Due to the complexity of potential energy and the large number of microstates,
analytical calculation of the partition function is not feasible except for very simple
systems with a few particles. Instead, in biomolecular simulations, we estimate the
partition function either by randomly sampling of the phase space or by propagating a
trajectory in the phase phase. In the next part I explain the concept of sampling as it is
presented in the statistics and statistical physics.
1.3.1 Sampling
In statistics, sampling is the process of selecting a subset of an original population.
A good sample is a statistical representation of the population properties, however as
the size of population increases appropriate sampling becomes more demanding. Due
to the size of phase space in biomolecular simulations, the importance of sampling has
been recognized. A good sample has two characteristics: it is unbiased, and it is large
enough to be precise. Later, I will explain more on those characteristics in the error
estimation section of this chapter.
15
1.3.2 Standard Sampling Techniques in Biomolecular Simulations
Molecular Dynamics and Monte Carlo are the two fundamental sampling methods in
the field of molecular modeling. Almost all other sampling methods are either inherited
or branched from one or both of these methods.
1.3.2.1 Molecular dynamics
Molecular Dynamics (MD) [7, 10, 11] simulation propagates a system in the phase
space by numerically integrating Newton’s equations. According to MD, the trajectory of
each particle (in the case of the Microcanonical ensemble) can be obtained by iteratively
solving Equations 1–4: F(X) = −∇U(X) = mV(t)
V(t) = X(t), (1–4)
where X and V are the vectors of position and velocity of the particle respectively. Then,
based on the ergodic hypothesis (see section 1.5.2), the average of an observable A
can computed by:
⟨A⟩ = limt→∞
1
M
M∑i=1
A(ti). (1–5)
where A(ti) is the value of A at time ti, also M is the total number of tis. The computational
problem with regular MD is that it may be trapped in local minima of the potential energy
surface. Such a trapping prevents the simulation from sampling the whole phase space
efficiently.
1.3.2.2 Monte Carlo
The Monte Carlo (MC) [7, 12, 13] technique was used to generate the first computer
simulation of a molecular system. In a Monte Carlo simulation, a new configuration is
introduced by applying a random change to the positions of the current configuration
of the system. Unlike the MD simulations there is no momentum contribution in
MC simulations and consequently there is no dynamics involves in the simulation.
16
Estimating the partition function for a system of N atoms using regular Monte Carlo
comprises the following steps:
1. Creating a configuration by randomly generating 3N Cartesian coordinates.
2. Calculating the potential energy, U(rN), and Boltzmann factor, exp(−βU(rN)), forthat configuration.
3. Repeating steps 1 and 2, M times (M >> 1) until adding new configuration doesnot change the average of observable A in the system,
⟨A⟩ = limM→∞
M∑i=1
Ai exp(−βU(rN))
M∑i=1
exp(−βU(rN)). (1–6)
Since many of the generated configurations may have no significant contribution to the
partition function, this is not a practical approach in systems with a large number of
particles. One way to overcome this barrier is use importance sampling techniques,
i.e. to add configurations to the sample based on their associated probability in the
partition. This idea was introduced by Metropolis et al[14]. Unlike in regular Monte
Carlo, in Metropolis Monte Carlo (MMC), one chooses the configurations based on their
probabilities and weights them evenly instead of choosing configurations randomly and
weighting them with proper probability afterwards. According to the MMC technique,
one needs to add a new step between steps 2 and 3 of regular MC in order to accept or
reject a new configuration. The probability of acceptance is
Pacc = min (1, exp (−β∆U)) , (1–7)
where ∆U is the difference between the potential energy of old and new configurations.
Considering the computational cost of MMC, although many of the randomly produced
configurations may be rejected, still one has to calculate the associated energy of all the
configurations to decide about acceptance or rejection.
17
1.4 Advanced Sampling Methods
In biomolecular systems with many degrees of freedom, there are large numbers
of local minima in the potential energy surface. Conventional sampling methods, such
as MC and MD, tend to be trapped in those minima, and consequently give inaccurate
thermodynamic averages. Advance sampling methods help to overcome this limitation.
There are many types of advanced sampling techniques. Here I explain two of the
more prominent categories of those: generalized ensemble methods and slow-growth
methods.
1.4.1 Generalized Ensemble Methods
Generalized Ensemble Methods[3, 15–17] overcome the problem of trapping in local
minima by changing the Boltzmann probability weights to non-Boltzmann weights such
that a random walk in potential energy space is realized.
1.4.1.1 Simulated tempering
In simulated tempering [15], temperature dynamically changes with time such
that the system accomplishes a random walk in temperature space. The probability of
visiting a microstate is proportional to the simulated tempering weight:
WST(E, T) = exp(−βE + a(T)), (1–8)
where a(T) is chosen such that the probability distribution of microstates in temperature
space becomes flat, i.e.:
PST(T) =
∫dEn(E)WST(E, T) = const, (1–9)
where n(E) is the density of energy states. In numerical simulations temperature axes is
limited to a list of M discrete values and the simulation can hop between those values.
The optimal values of a(T) should be determined by iteration of trial simulations. The
following is a summary of the ST algorithm:
1. Running a canonical MD or MC simulation at temperature Ti for a certain steps.
18
2. Changing Ti to one of its earliest neighbors in the temperature list with transitionprobability of:
π(Ti → Ti±1) = min
(1,
WST(E, Ti±1)
WST(E, Ti),
), (1–10)
which is based on MMC criteia.
1.4.1.2 Multicanonical algorithm
In the Multicanonical Algorithm (MUCA) [17, 18], a flat potential energy distribution
is achieved by weighting each microstate using a non-Boltzmann weight factor, WMUCA,
which implies:
p(E) ∝ n(E)WMUCA(E) = const. (1–11)
This induces a random walk in the energy space of the system, which prevents a
simulation from trapping in local minima of the potential energy surface of the system.
Similar to the ST method, the density of states is not a priori known and the MUCA
weight factor should be determined by iteratively running short trial simulations[17, 19].
One can perform MUCA using both MD and MC methods. Moreover the average
of physical quantity A at temperature T can be calculated using single reweighting
techniques:
⟨A⟩T =
∑i
A(qi)W−1MUCA(E(qi)) exp (−βE(qi))∑
i
W−1MUCA(E(qi)) exp (−βE(qi))
(1–12)
where qi is the ith configuration.
1.4.1.3 Wang-Landau sampling
The Wang-Landau algorithm[16] is a MMC technique for estimating the density
of states. It can be used as a complementary tool for determining the weight factor in
MUCA. In this method, we initially set all n(E) = 1 and the probability of jumping from an
19
energy state Ei to Ej is
p(Ei → Ej) = min
(1,
n(Ej)
n(Ei)
). (1–13)
which induced a random walk in enery space of the system. Each time the current
energy density of state updates as following:
n(E)← n(E)f, (1–14)
where f is the modification factor. Typically a simulation begins with f = e ≃ 2.71828.
During the random walk, we accumulate a histogram H(E), which is the number of
visits at each energy level E. When the histogram becomes flat, we reset the density of
states, update the modification factor f ←√f , and restart the whole process from the
beginning. We can continue this scheme until we reach desired accuracy in estimating
of the density of states.
1.4.1.4 Replica exchange
The Replica Exchange (RE) method, which originally was proposed by Swendsen[1,
20], is one of the most successful methods in the field of molecular simulations. This
method has two advantages: First, unlike ST and MUCA, in RE, the weight factor is
known a priori. Second, RE can intuitively be combined with many other enhanced
sampling methods[21–24]. In this technique, M independent copies (replicas) of the
system are treated by MD (i.e., REMD) or MC (i.e., REMC) simultaneously, and they
ask to exchange their configurations after every certain numbers of steps of simulation.
By exchanging configurations between replicas, a random walk in the replicas’ ladder
space is realized. Based on the type of RE, replicas can span temperature space
(Temperature RE), Hamiltonian space (Hamiltonian RE), or both (Multi Dimensional RE).
The core of the exchange criteria is based on imposing a detailed balance equation to
20
the generalized ensemble:
w(X→ X)℘(X) = w(X→ X)℘(X), (1–15)
where ℘(X) and w(X → X) are the probability of being at state X and the probability of
transition from X to X, respectively. X and X are generalized ensembles before and after
the exchange, i.e.,
X =
T1 ... Ti Tj ... TM
H1 ... Hi Hj ... HM
q1 ... qi qj ... qM
, X =
T1 ... Tj Ti ... TM
H1 ... Hj Hi ... HM
q1 ... qi qj ... qM
(1–16)
Here Hi,Ti, and qi represent the Hamiltonian, temperature, and configuration of replica
i just before exchange. Because all replicas are independent, the probability of the
system being at the generalized state of X is ℘(X) =∏M
i=1 pi(Ti, Hi, qi), where
pi(Ti, Hi, qi) is the probability that replica i, with Hamiltonian Hi and temperature of
Ti is found with the conformation state of qi. In the case of canonical ensemble
pi(Ti, Hi, qi) ∝ exp(−βiHi(qi)), (1–17)
where βi =1
kBTi. Then the detailed balance equation can be written as:
w(X→ X)
w(X→ X)=
pi(Tj, Hj, qi)pj(Ti, Hi, qj)
pi(Ti, Hi, qi)pj(Tj, Hj, qj)=
exp(−βjHj(qi)) exp(−βiHi(qj))
exp(−βiHi(qi)) exp(−βjHj(qj))= exp(−∆),
(1–18)
where
∆ = [(βjHj(qi) + βiHi(qj))− (βiHi(qi) + βjHj(qj))] . (1–19)
Then using the MMC crieria, the probability of transition from X to X can be calculated
as:
w(X→ X) = min (1, exp(−∆)) . (1–20)
21
In the case of TRE, where all Hamiltonians are the same, Equation 1–19 can be
simplified to:
∆ = [(βj − βi)(H(qj)− H(qi))] , (1–21)
and in the case of HRE, where all temperatures are equal, it can be written as:
∆ = β [Hj(qi) + Hi(qj)− Hi(qi)− Hj(qj)] . (1–22)
1.4.1.5 Umbrella sampling
The Umbrella Sampling method, which was introduced by Torrie and Valleau[25]
in 1977, is one of the major techniques for calculating the PMF along preset reaction
coordinates. In this method, we restrain the system at different part/parts of the reaction
coordinate by adding single/multiple bias potential/potentials. The bias is an additional
potential energy term, which restrains the system to the reaction coordinate. Usually, the
bias is a quadratic function of the reaction coordinate:
B(ξ) = k(ξ − ξ0)2, (1–23)
where k, ξ, and ξ0 are the bias strength, the order parameter and the bias center on
the reaction coordinate and configuration respectively. In order to extract an estimation
of the PMF as a function of a reaction coordinate, it is necessary to remove the effect
of bias using post-processing methods. Many methods have been proposed in the
literature; among those, the Weighted Histogram Analysis Method (WHAM)[26] and
Umbrella Integration (UI)[27, 28] are the most promising.
1.4.2 Slow-Growth Methods
Slow-growth methods are used to compute the PMF between two given states (e.g.,
A and B) of a system by summing the free energy differences between the intermediate
22
steps. Here the potential energy for intermediate stages is defined as:
U(λ) = UA + λ(UB − UA), (1–24)
here λ is called the coupling parameter, U(0) = UA , and U(1) = UB. In slow-growth
techniques, intermediate states i.e. 0 < λ < 1, may not have any physical meaning.
Consequently, for the case of the canonical ensemble, the Helmholtz free energy at the
state of λ can be written as:
A(N,V,T,λ) =−1β
ln(Z(N,V,T,λ)), (1–25)
where Z(N,V,T,λ) =∑
i exp(−βUi(λ)) is the partition function for the state of λ. Here
I explain Thermodynamics Integration (TI) and Free Energy Perturbation (FEP), which
are among the prevalent slow-growth methods. One of the applications of this technique
is the calculation of pKas of different species in proteins[29]. In this type of calculation,
one of the two end states (i.e., λ = 0 or 1) characterizes the protonated state, and the
other represents the deprotonated state or vice versa. The intermediate λs correspond
to hybrid protonated and deprotonated states.
1.4.2.1 Thermodynamic integration
According to the TI technique, the free energy difference between states A and B is:
∆A(A→ B) =
∫ 1
0
dλ∂A
∂λ. (1–26)
Using Equation 1–25 this can be rewritten as:
∆A(A→ B) =
∫ 1
0
dλ−1βZ
∂Z
∂λ=
∫ 1
0
dλ1
Z
∑i
exp(−βUi(λ))∂Ui(λ)
∂λ=
∫ 1
0
dλ⟨∂U(λ)∂λ⟩λ,
(1–27)
where the bracket average represent an ensemble average generated at λ. In practice,
one has to break the integration to the sum of infinitesimally different intermediate
23
states. Moreover, the integration path should be reversible, i.e., changes in λ should be
small enough, and the system should be relaxed at each λ value.
1.4.2.2 Free energy perturbation
The Free energy perturbation method, which was introduced by Zwanzig in
1954[30], is another frequently used free energy difference method. Considering
the canonical ensemble, the Helmholtz free energy difference between states A and B
can be expressed as:
∆A(A→ B) =−1β
lnZB
ZA
=−1β
ln∑ exp(−βUB)
ZA
=−1β
ln∑ exp(−βUA)
ZA
exp(−β(UB − UA))
=−1β
ln ⟨exp(−β(UB − UA))⟩A.
(1–28)
During the derivation of Equation 1–28, we implicitly assumed the number of energy
microstates in states A and B to be equal, but this is only true when two states are
infinitesimally close. One can use the Equation 1–24 to adapt the Equation 1–28 for
computing the free energy difference between distinct states in the system:
∆A(A→ B) =−1β
M−1∑i=1
ln ⟨exp(−β(Uλi+1− Uλi
))⟩λi, (1–29)
where M is the number of discrete steps which has been used for moving from state A
to state B.
1.5 Convergence, Error Estimation, and Sampling Quality in BiomolecularSimulations
1.5.1 Error in Computational Methods
There are two types of errors in estimating a variable using computational methods,
systematic and random. Random or statistical errors, which are associated with
precision of measurements, are fluctuations in the measured data due to imperfect
sampling. In order to estimate the random error, one needs to repeat the simulation
24
many times and calculate the variance of the measured variable. In contrast, systematic
errors or biases are reproducible inaccuracies, which usually come from the deficiency
of energy and force computation methods. The bias can be defined as a difference
between the correct value and the measured value. The estimation of bias is complicated,
since the correct value of a variable is not a priori known. To overcome this problem,
typically the bias is measured with respect to a more sophisticated model (e.g., bias of
a force field method with respect to a quantum method). On the other hand, random
error is associated with the quality of sampling. In the next section I describe the ergodic
theory, which is the base of many sampling quality measurement methods.
1.5.2 Ergodicity and Convergence
Ergodic theory, which originated from Boltzmann’s work in statistical physics,
describes the behavior of a dynamical system when it runs for a long time. According
to this hypothesis, a system eventually visits all points in its phase space, if it is allowed
to run for enough time. One of the consequences of ergodic theory asserts that, under
certain conditions, the time average of a function over a dynamics trajectory is equal to
the space average of that function, i.e.:
⟨A⟩ =∫p
∫q
p(q, p))dp dq = limt→∞
1
t
∫ t0+t
t0
A(τ)dτ , (1–30)
where p(q, p) is the probability of the system having at configuration q and the
momentum of p, moreover t0 is an arbitrary origin on the time coordinate.
The ergodic theory is closely related to the concept of convergence. In the context
of molecular simulations, a simulation is considered converged when the collected
set (sampled set) is a precise representation of the original ensemble. Moreover, a
system is called semi-ergodic, when some states are not accessible during a reasonable
simulation time, so the system will not converge easily. Since the original ensemble is
not feasible, full convergence is not practicable either. In other words, one can approach
the ideal ensemble, but cannot achieve it. In the next parts, I discuss some tests that
25
quantify the degree of confidence about a simulation convergence. We have to note that
those assessments only describe necessary assurances, but not the sufficient ones.
1.5.3 Sampling Quality Checks
One of the most important questions in sampling methods is this: when have
we collected enough samples? To answer this question, we need to quantify the
convergence rate and the sampling quality in a simulation. Here I describe the most
employed convergence analysis tools in computational biophysics and biochemistry.
1.5.3.1 Root mean square deviation analysis
Root Mean Square Deviation (RMSD) of the configurations measures the average
movements of particles in a snapshot from (usually) the initial snapshot. For a system
with N particles, It is defined as :
RMSDt =
√∑Ni=1 |Xi(t)− X0
i |2N
(1–31)
where |Xi(t)− X0i | is the spatial displacement of particle ith between time t and time 0.
One way to track the convergence rate is to calculate the cumulative RMSD average
as function of time:
⟨RMSD⟩t =∑
t RMSDt
n. (1–32)
where n is the number of snapshots from beginning to time t. This method is useful
for tracking the folding state of protein, since in a folded state the protein fluctuates
around the average configuration. The definition above is not unique and there are other
variations of RMSD definitions[31–35].
1.5.3.2 Root mean square deviation clustering
Clustering is the process of organizing a set of objects into groups whose members
are related in some way. Clustering is a part of many statistical analyses. However,
considering the long list of those methods is beyond the scope of this dissertation.
Among them, Root mean square deviation clustering is frequently used in the context
26
of molecular simulations. This method can be used for dividing an ensemble into
sets of self-similar structures. Daura et al[36] adopted this method for assessing the
convergence rate in the simulation. According to this method, a significant decrease
in the rate of discovery of new clusters during the simulation is a sign of convergence.
However a closer investigation reveals that this condition is necessary but not sufficient.
For example, trapping in energy minima can significantly decrease the rate of cluster
discovery in a simulation, although the simulation is not converged.
1.5.3.3 Block averaging
Block Averaging (BA), as proposed by Flyvbjerg and Petersen[37], is a method
for examining the quality of sampling in a correlated simulation. This method can be
described as following: A trajectory with time length of τ = M.n snapshots is split in to M
different block of size n, starting with small n, e.g., n = 1. The mean of observable A is
calculated for each block. Then one can calculate the Blocked Standard Error (BSE) as:
BSE(A, n) = σn/√M, (1–33)
where σn is the variance among M blocked averages. For small values of n (i.e., small
blocks) blocks are more correlated, So BSE underestimates the standard error. As the
value of n increases (or M decreases) the correlation decreases and BSE converges
to a value. Plotting the BSE(A, n) vs. n comprises a signal of both statistical error
convergence and decorrelation of A.
1.5.3.4 Principal component analysis
Principal Component Analysis (PCA) [? ] is a technique for calculating the
large-scale characteristic motions from a simulation trajectory. In order to perform
PCA, one needs to create the 3N × 3N (where N is the number of atoms) covariance
matrix as:
Cij = ⟨(xi − xi) (xj − xj)⟩ (1–34)
27
and compute the eigenvalues and eigenvectors of the matrix. Where xi represents
ith degree of freedom in the configuration space. Each eigenvalue represents the
mean square deviation along the corresponding eigenvector. The modes with higher
eigenvalues signify the main directions of motion in the system. Another application of
PCA is for measuring the degree of similarity in the variations of two trajectories[38].
Also, this can be used as a test of convergence in a single long trajectory by dividing the
long trajectory into trajectories with smaller lengths.
1.5.3.5 Kullback-Leibler divergence
Kullback-Leibler divergence[39, 40] or relative entropy is defined as:
DK−L(P||Q) =∑i
P(i) ln
(P(i)
Q(i)
), (1–35)
where P and Q are the probability distributions of a random variable. It is the average,
over the distribution P, of the logarithmic difference between the probabilities P and
Q.It measures the degree to which P is distinguishable from Q. DK−L is non-negative
quantity and a small value of it indicates that P and Q are highly overlapped. This metric
can be used as a measure of convergence rate, where Q is a target (or reference)
distribution and P is changing with time.
1.6 Outline of My Research
During my Ph.D studies, I have been mostly focused on developing and testing
enhanced sampling methods and mainly Hamiltonian Replica Exchange Molecular
Dynamics (HREMD).
My first project, which is described in the next chapter, was about developing
Replica Exchange Free Energy Perturbation (REFEP) method, which is a combination
of the Free Energy Perturbation (FEP) and HREMD[22]. We demonstrated that HREMD
method not only improves convergence in alchemical free energy calculations, but also
can be used to compute free energy differences directly via the FEP algorithm. We
showed a direct mapping between the HREMD and the usual FEP equations, which are
28
then used directly to compute free energies. We tested REFEP on predicting the pKa
value of the buried Asp26 in thioredoxin. We compared the results of REFEP with TI and
regular FEP simulations. REFEP calculations converged faster than those from TI and
regular FEP simulations. The final pKa value predicted from HREMD simulation was only
0.4 pKa units above the experimental value. Briefly, we showed REFEP algorithm not
only improves conformational sampling, but also improves the convergence rate of free
energy simulations.
My next project, i.e., the third chapter, was a demonstration of the capabilities of
REFEP in estimating the pKa of complicated proteins[41]. According to the experimental
measurements, the pKa value of Glutamate 66 (GLU66) in a hyperstable mutant of
staphylococcal nuclease displays a large shift, roughly 4.6 pKa units, relative to its
normal value in water, as measured in the lab of Moreno et al [83, 84, 92–98]. In order
to reproduce the large experimental shift using a single structure, continuum solvent
and computational methods, an internal dielectric constant around 10 is necessary. The
physical reason for this is not yet understood, but hypotheses have been produced by
Moreno et al [91, 97, 99, 104] regarding solvent penetration, protein reorganization, etc.
We aimed to resolve this inconsistency between experimental and continuum methods
by introducing a four-state thermodynamic cycle that couples conformational states
with protonation states of GLU66. We proposed that the experimental methods (which
are mostly sensitive to configurational changes) measure the equilibrium constant
between the two configurational states instead of the two protonation states. We used
REFEP method in implicit solvent to calculate the pKa value of GLU66 for each of the
configurational states as well as the mixed configuration. The results are in almost
perfect agreement with the experiments of Moreno et al.
In my third research project I developed and validated a pH-Replica Exchange
Molecular Dynamics (pH-REMD) method[24]. This method improves the coupling
between conformational and protonation sampling. Under a HREMD setup, conformations
29
are swapped between two neighboring replicas, each having different pHs. We applied
pH-REMD to a series of model compounds, a terminally charged ADFDA pentapeptide,
and a heptapeptide derived from the ovomucoid third domain (OMTKY3). In all of those
systems, the predicted pKa values by pH-REMD were very close to the experimental
values and almost identical to the ones obtained by constant pH molecular dynamics
(CpH MD).
In the last year of my PHD, I have focused on optimizing HREMD methods. The
goal is finding the best position for replicas in order to maximize the round trip between
extremum positions on replica ladder. We developed, validated, and tested a method
for estimating the probability of exchange between neighboring replicas in Umbrella
Sampling Replica Exchange MD (USRE)[42]. We use information from very short
umbrella runs, needing only a handful of windows. We designed a multi dimensional
scoring function to optimize the set of replicas (windows). By maximizing the scoring
function, we enforce the same exchange acceptance for all neighbor replica pairs. We
found having equal exchange acceptance between pairs increases the number of round
trips and improves the efficiency of sampling. A description of this method can be found
in chapter 5.
30
CHAPTER 2COMPUTING ALCHEMICAL FREE ENERGY DIFFERENCES WITH HAMILTONIAN
REPLICA EXCHANGE MOLECULAR DYNAMICS (HREMD) SIMULATIONS
2.1 Literature Review
Free energy, especially the free energy difference between two states, is a crucial
quantity in the study of chemical and biological systems.[43] Knowledge of the free
energy differences can help us understand the behaviors of such systems. For example,
the free energy of binding is one of the criteria used to evaluate the performance of
drugs.[44] Therefore, one important aspect of molecular modeling is to yield accurate
free energy differences efficiently. Many free energy calculation methodologies (such as
free energy perturbation,[30] thermodynamic integration,[45] umbrella sampling,[25, 46,
47] and Jarzynski’s equality [48] as well as analysis techniques (such as the weighted
histogram analysis method[49] and Bennett acceptance ratio method [50, 51] have been
developed to achieve this goal. In general, free energy calculations could be divided
into alchemical free energy and conformational free energy calculations. The alchemical
free energy calculations are often employed when studying the free energy differences
of processes that involve changes in noncovalent interactions. In an alchemical free
energy simulation, a nonphysical reaction coordinate λ is generally adopted in to
connect the initial and final states. This reaction coordinate is usually expressed as
an interpolation of the initial and final states. Thus, an alchemical process is achieved
through a series of intermediate states having no direct physical meaning. Since the free
energy difference between two states is a state function, the actual choice of coordinate
cannot, in the limit of infinite sampling, affect the results. Free energy perturbation (FEP)
Reprinted with permission from Meng, Y.; Dashti, D. S.; Roitberg, A. E. J. Chem.Theory Comput. 2011, 7, 2721–2727.
31
and thermodynamic integration (TI) are two common methodologies that are utilized in
alchemical free energy computations.
One important issue in alchemical free energy calculations is the convergence of
the free energy difference versus computational cost. Such convergence is particularly
difficult in systems involving slow structural transition or large environmental reorganization
as λ changes.[21, 52, 53] Therefore, conformational sampling is crucial in alchemical
free energy calculations. Enhanced sampling methods, such as replica exchange
molecular dynamics (REMD),[2] orthogonal space random walk (OSRW),[53] and
accelerated molecular dynamics (AMD)[54] have been applied to free energy simulations
in order to accelerate conformational sampling and, in turn, to yield accurate and
converged free energy differences. Among the enhanced sampling methodologies, the
REMD method is of particular interest because the weight of each state is a priori known
(Boltzmann factor).
Both temperature-based and Hamiltonian-based REMD have been applied to
alchemical free energy calculations. Woods et al.[21] and Rick[60] have combined the
temperature-based REMD with TI calculation. A temperature-based REMD simulation
is conducted at each state along the reaction coordinate. Woods et al.[21] have also
applied the HREMD methodology to FEP and TI calculations. Each replica in the
HREMD simulation represents a state along the reaction coordinate λ, and a periodic
swap in λ is attempted. Relative solvation free energy of water and methane as well
as the relative binding free energies of halides to calis pyrrole have been calculated
in this way[21]. The Yang group has developed a dual-topology alchemical HREMD
(DTA-HREM) method[61]. Their method was tested on the free energy of mutating
an asparagine amino acid (with two ends blocked) to leucine. More recently, the
Roux group coupled the FEP methodology with the distributed replica technique
(REPDSTR)[62, 63]. An additional acceleration in the sampling of the side-chain
dihedral angle was also incorporated when Jiang and Roux utilized the FEP/HREMD
32
method to study the absolute binding free energy of p-xylene to the T4 lysozyme L99A
mutant [63]. In all of those studies, the conformational sampling and convergence of
free energy computations showed significant improvement when the REMD method
was applied. The protocol presented here accelerates convergence but, of course,
does not solve known problems in the field related to enhanced sampling of coordinates
orthogonal to λ space, which would hamper many of the current methods.
In this chapter, we will demonstrate that FEP is actually already incorporated in the
HREMD method in an elegant and formal way. The REFEP method is shown to be not
only an enhanced sampling method but also a free energy calculation algorithm. We will
apply the REFEP method to the pKa prediction of thioredoxin Asp26. The experimental
pKa value of 7.5 has been shown to be one of the largest shifted from the intrinsic pKa
value[64, 65] and, hence, makes it an interesting case to be studied theoretically. TI and
FEP (regular molecular dynamics for conformational sampling) alchemical free energy
simulations have been conducted in order to compare with REFEP simulations. A very
accurate theoretical pKa value is obtained from REFEP simulations. The convergence
of the free energy difference and pKa value is achieved in REFEP simulations much
faster than that in the FEP and TI simulations. The advantage and simplicity of using the
HREMD simulation to compute the alchemical free energy difference is clearly shown.
2.2 Theory and Method
2.2.1 Free Energy Perturbation(FEP)
The FEP method, which was initially introduced by Zwanzig in 1954[30], is a well
established method and is considered the most frequently employed methodology
in alchemical free energy calculations[52]. The details of the FEP, as well as the TI,
methodology and its applications have been extensively reviewed[52, 66–69]. Therefore,
only a very brief description of the FEP and TI methods will be given here. Consider two
states (1 and 2) of a system in the canonical (NVT) ensemble, and their corresponding
Helmholtz free energies A1 and A2. The Helmholtz free energy difference between two
33
states can be expressed as
∆A1→2 = −kBT ln⟨exp([U2(q)− U1(q)]
kBT
)⟩1. (2–1)
Here, kB is the Boltzmann constant, T is the temperature, and q is the molecular
structure. U1 and U2 are the potential energies of states 1 and 2, respectively.
The bracket with subscript 1 stands for the average calculated over the structural
ensemble generated by state 1. In order to compute ∆A1→2, one simulation of state
1 is performed. Once a configuration q is taken, the potential energy difference at
configuration q is computed. The ensemble average, which is ⟨exp(
[U2(q)−U1(q)]kBT
)⟩1,
can be calculated easily, and hence, ∆A1→2 is obtained. Although the Helmholtz free
energies are utilized here, Equation 2–1 can be extended to an isothermal-isobaric
(NPT) ensemble and to the Gibbs free energy in the same manner.
When the fluctuations in ∆U in Equation 2–1 are too large, FEP calculations are
notoriously hard to converge. The convergence of the FEP calculation will be poor if
the overlap in phase space between the two states is small. In order to compute the
free energy difference between two states that are very different, intermediate states
mixing the two end points are adopted in such a way that the differences between
neighbors can be treated as perturbations. A frequently employed method to generate
intermediate states is to interpolate potential energy functions linearly, as shown in
Equation 2–2. In Equation 2–2, U1 and U2 are the potential energy functions of states
1 and 2, respectively. Free energy differences between neighboring states are then
computed. The sum of individual free energy differences will be the targeted free energy
difference between states 1 and 2 (Equation 2–3). There are many ways of executing
FEP calculations involving intermediate states. The double-ended, double-wide[67, 70],
and overlap sampling algorithms[71] are among the most popular ones. A thorough
description of different algorithms and their performance can be found in a recent review
34
by Jorgensen and Thomas[67].
U(λ) = U1 + λ(U2 − U1), (2–2)
∆A1→2 = −kBT∑i
ln⟨exp([U(λi+1)− U(λi)]
kBT
)⟩i. (2–3)
In practice, computing ∆A1→2 (forward free energy difference) is equally easy (or hard)
as computing ∆A2→1 (backward free energy difference), and one is exactly the opposite
of the other in principle. Evaluation of forward and backward free energy differences
provides an indication of convergence. Furthermore, the potential energy differences
generated from both directions can be utilized to reduce statistical error. The Bennett
acceptance ratio (BAR) method is a frequently employed scheme to improve the
precision of a free energy estimator[50–52].
2.2.2 Thermodynamic Integration
Another way of writing the free energy difference between two states 1 and 2 is
∆A1→2 = −kBT∑i
ln⟨exp([U(λi+1)− U(λi)]
kBT
)⟩i. (2–4)
Here, λ is a reaction coordinate connecting states 1 and 2, and U is the potential energy
of a state along the reaction coordinate. The bracket represents an ensemble average
generated at a value of λ. The integration is often evaluated numerically via trapezoidal
rule or Gaussian quadrature. If U(λ) is constructed as in Equation 2–2, the derivative of
U(λ) with respect to λ is
∂U(λ)
∂λ= U2 − U1. (2–5)
And the free energy difference between states 1 and 2 can be expressed as
∆A1→2 =
1∫0
⟨U2 − U1⟩λdλ (2–6)
35
Hence, the ensemble average of the potential energy gap between states 1 and 2 at
each λ value is needed in a TI calculation. In this chapter, we use the term TI to refer to
constrained TI, in which the value of λ is not allowed to change at each window.
2.2.3 Hamiltonian Replica Exchange Molecular Dynamics (HREMD)
The original REMD method utilizes replicas having different temperatures (TREMD).
Replicas at high temperatures overcome potential energy barriers more easily than
those at low temperatures. Another way to overcome potential energy barriers is simply
to change the potential energy surface to reduce potential energy barriers. In the
HREMD algorithm, replicas differ in their Hamiltonians but have the same temperature.
Regular MD is performed, and an exchange of configurations between two neighboring
replicas is attempted periodically.
Figure 2-1 demonstrates the HREMD algorithm and the free energy computation in
an HREMD simulation. Let us consider two replicas 1 and 2 with corresponding potential
energies U1 and U2. By employing the detailed balance condition and Boltzmann weight
of each molecular structure, the transition probability can be written as
w(q1 → q2) = min 1, exp [−(U1(q2) + U2(q1)− U1(q1)− U2(q2))/kBT] (2–7)
where q1 and q2 are the molecular structures of replicas 1 and 2 before an exchange
attempt, respectively. A Monte Carlo Metropolis criterion[14] is used to evaluate whether
the attempted swap of structures between two replicas should be accepted or not.
Equation 2–7 can be regrouped as
w(q1 → q2) = min 1, exp− [(U2(q1)− U1(q1) + U1(q2)− U2(q2))/kBT (2–8)
When comparing the exponential terms in Equation 2–1 and 2–8, it is clear that
Equation 2–8 incorporates all information necessary for a FEP calculation. U2(q1) −
U1(q1) is the potential energy difference computed on the basis of the structural
ensemble generated by U1, while U1(q2) − U2(q2) is the potential energy difference
36
Figure 2-1. Diagrams displaying the HREMD exchange algorithm and free energycalculation. (A) Exchange attempt orders. Replicas connected by a curveare neighbors, and attempts are made to exchange molecular congurations(q). (B) Free energy calculations in the HREMD method. Each replica hastwo free energy differences: Aup and ∆Adown from its attempting neighborform a pair and are computed simultaneously, while ∆Adown and ∆Adown fromits attempting neighbor form the other pair. In exchange attempts (regardlessif the attempts are accepted or rejected), two pairs of free energy differencesare computed in an alternating fashion utilizing Equation 2–1.
computed on the basis of the structural ensembles generated by U2. Every time the
transition probability is computed, those potential energy differences can be utilized to
compute the ensemble average shown in Equation 2–1. Therefore, ∆A1→2 and ∆A2→1
can be computed on-the-fly utilizing the double-ended scheme. The ensemble average
in Equation 2–1 is computed regardless of whether an exchange attempt is accepted
or rejected. When employing the HREMD method to improve conformational sampling
in the study of alchemical changes, HREMD simulations are able not only to enhance
conformational sampling but also to yield the free energy difference directly. In fact,
37
a regular FEP calculation can be thought of as an HREMD calculation in which no
exchanges are allowed between replicas.
In practice, as shown in Figure 2-1, there are two free energy difference calculations
(∆Aup and ∆Adown) continuously associated with each replica. Take replica 1 as an
example: ∆Aup = ∆A1→2 while ∆Adown = ∆A2→1. In principle, when converged, ∆A1,up
should be equal to the negative of ∆A2,down:
∆A1,up = −kBT ln⟨exp([U2 − U1]
kBT
)⟩1 = −∆A2,down
= kBT ln⟨exp([U1 − U2]
kBT
)⟩2
(2–9)
Any difference (except for the sign) between the two is an indication of error or lack of
convergence.
Convergence also was gauged by the time dependence of the predicted free energy
differences, computing ∆G versus simulation length. This provides an asymptotically
unbiased estimator for ∆G, thus all methods presented here must eventually reach the
same final value (within error bars). REFEP is presented in this chapter as showing
faster convergence toward the final value.
2.2.4 Simulation Details
Accurately determining the pKa values of ionizable residues, especially those
with large shifts from intrinsic pKa values, is of great interest both experimentally and
computationally[64, 65, 72]. Here, the pKa calculation of Asp26 in thioredoxin has been
selected as a test case in order to compare the performance of alchemical free energy
simulations. Asp26 has been found deeply buried in thioredoxin and possesses one of
the largest pKa shifts among protein carboxylic groups[64, 65]. Following the protocol
employed in the paper of Simonson et al [72], the thermodynamic cycle utilized to
compute the pKa value of an ionizable residue is given in Figure 2-2. As it can be seen
there, the use of a model compound as an auxiliary leg in the thermodynamic cycle
makes ∆G3 (proton to proton) equal to zero. Essentially, the pKa shift relative to the
38
intrinsic value (pKa,model) is computed as
pKa(protein) = pKa(model) +1
2.303kBT
[∆G(proteinAH→ proteinA−)−∆G(AH→ A−)
](2–10)
where ∆G(proteinAH → proteinA−) and ∆G(AH → A−) are the free energy differences
between protonated and deprotonated aspartic acid in the protein environment and in
aqueous solution, respectively. Alchemical free energy simulations were performed in
order to yield those two terms. In Equation 2–10, the Gibbs free energy differences are
used because experiments determining pKa values generally are conducted under an
isobaric-isothermal condition.
Figure 2-2. Thermodynamic cycle used to compute the pKa shift. Both acid dissociationreactions occur in aqueous solution. The protein-AH represents the ionizableresidue in a protein environment. The AH represents the model compoundwhich is usually the same ionizable residue with capped terminii. In practice,a proton does not disappear but instead becomes a dummy atom. Theproton still has its position and velocity. The bonded interactions involvingthe proton are still effective. However, there are no nonbonded interactionsfor that proton. The change in the ionization state is reflected by changes ofpartial charges in the ionizable residue.
Aspartic acid dipeptide in implicit water solvent was taken as the model compound
with a pKa value taken as 4.0[73]. The oxidized form of thioredoxin (PDB code
2TRX)[74] in implicit water was used in our simulation. Changes in ionization were
represented by changes in the partial charges of the aspartic acid side chain (ASH→ASP
39
in the AMBER terminology). Since the van der Waals radius of the proton in aspartic
acid is zero for both protonated and deprotonated species, the free energy difference
only contains the electrostatic interactions.
Three types of free energy simulations were performed for both the model
compound and the protein: TI (forward and backward), HREMD-FEP (REFEP), and
regular FEP simulations. Our regular FEP simulations were carried out via HREMD
simulations but with all exchange attempts rejected. Comparing the pKa prediction
and free energy convergence from FEP and REFEP simulations will directly indicate
the effect of the enhanced conformational sampling due to the exchanges. Linear
interpolation of point charges was carried out in order to assign side chain charges for
intermediate states. A seven-point Gaussian quadrature was selected to compute total
free energy difference for TI calculations. Therefore, eight λ values (one end point is
needed in either direction) were utilized in the TI simulation. Due to the implementation
of the TI algorithm in AMBER [75], 16 replicas were utilized to ensure the same amount
of simulation time for all free energy simulations. A simulation time of 5 ns was used
for each λ value and for each replica in the study of the model compound, while for
thioredoxin, we used 4 ns runs. Structural swaps between neighboring replicas were
attempted every 2 ps (1000 MD steps). No particular attempt was made in this work to
optimize the number or location of the replicas, nor the exchange attempt frequency.
All simulations were done using the AMBER 10 molecular simulation suite[75],
locally modified to add HREMD/REFEP capabilities. The AMBER ff99SB force field [76]
was utilized in all of the simulations. The SHAKE algorithm[77] was used to constrain
the bonds connecting hydrogen atoms with heavy atoms in all of the simulations,
which allowed the use of a 2 fs time step. The OBC (Onufriev, Bashford, and Case)
generalized Born implicit solvent model (igb = 5 in the AMBER terminology)[78] was
used to model the water environment in all of our calculations. The cutoff for nonbonded
interaction and the Born radii was set to 99 A. This value is larger than the dimension of
40
both systems. Langevin dynamics was employed in order to maintain the temperature at
300 K, using a friction coefficient of 3.0 ps−1.
2.3 Results and Discussions
2.3.1 Acceptance Ratio of HREMD Simulations
The accuracy of FEP depends on the overlaps between phase spaces, which can
be measured as overlaps between potential energy difference distributions[52]. The
acceptance ratio in an HREMD simulation is an indication of the overlap between two
potential energy difference distributions[61]. Therefore, it could be utilized to monitor
the convergence of free energy calculation qualitatively. In our study, large acceptance
ratios were observed in both the model compound and protein HREMD simulation.
The acceptance ratio between two neighbors ranged from 0.7 to 0.9 in all HREMD
simulations. Those large acceptance ratios indicate that the overlap in phase space is
large.
2.3.2 Aspartic Acid Model Compound Study
The free energy differences on the right-hand side of Equation 2–10 were
calculated as described in the Theory and Method section. The cumulative average
free energy difference as a function of time is reported here. Figure 2-3A shows the
∆G(AH → A−) from TI, HREMD, and FEP simulations (as mentioned before, a FEP
simulation has been performed by rejecting all exchange attempts in an HREMD
simulation). The differences between forward and backward ∆G(AH → A−) are shown
in Figure 2-3B. A converged alchemical free energy simulation should generate the
same forward and backward free energy numerically (except for an opposite sign). Any
nonzero value is an indication of free energy not converged.
For a simple system such as aspartic acid in implicit water, 5 ns of simulation time
was long enough for ∆G(AH → A−) to stabilize in all three alchemical free energy
simulations, as shown in Figure 2-3A. The forward and backward ∆G(AH → A−) at
the end of each free energy calculation and the corresponding error bars are listed
41
Figure 2-3. (A) Cumulative average free energy differences between protonated anddeprotonated aspartic acid in the model compound (∆G(AH→ A−)). (B)The differences between forward and backward ∆G(AH→ A−). (C)Cumulative average free energy differences between protonated anddeprotonated Asp26 in thioredoxin (∆G(proteinAH→ proteinA−)). (D) Thedifferences between forward and backward(∆G(proteinAH→ proteinA−)).
42
Table 2-1. Free energy difference (in Kcal/mol) between protonated and deprotonatedAspartic acids obtained from TI, REFEP, and FEP alchemical free energysimulations
TI REFEP FEP
ASP modelforward -59.43± 0.06 -59.69 ± 0.05 -59.84 ± 0.06backward -59.56± 0.06 -59.66 ± 0.05 -59.72 ± 0.06average -59.50 ± 0.08 -59.68 ± 0.08 -59.78 ± 0.08
Asp26 in thioredoxinforward -54.35 ± 0.61 -54.29 ± 0.17 -54.23 ± 0.56backward -55.82 ± 0.39 -54.24 ± 0.14 -53.84 ± 0.56average -55.09 ± 0.72 -54.27 ± 0.22 -54.04 ± 0.79
∆G differenceforward 5.08 ± 0.61 5.40 ± 0.18 5.61 ± 0.56backward 3.74 ± 0.39 5.42 ± 0.15 5.88 ± 0.56average 4.41 ± 0.72 5.41 ± 0.23 5.74 ± 0.79
predicted pKa,protein
forward 7.7 ± 0.4 7.9 ± 0.1 8.1 ± 0.4backward 6.7 ± 0.3 7.9 ± 0.1 8.3 ± 0.4average 7.2 ± 0.5 7.9 ± 0.2 8.2 ± 0.6
in Table 2-1. The forward and backward free energy differences are the same (within
error bars) for both REFEP and FEP simulations. However, the TI simulations failed
to do that, although the difference was very small (the difference between forward
and backward ∆G(AH → A−) was only 0.13Kcal/mol). The average of forward and
backward ∆G(AH → A−) was taken as the final value of ∆G(AH → A−) for the model
compound and is also reported in Table 2-1. Clearly, as shown in Figure 2-3B, the
REFEP simulations have converged much faster than the FEP calculations did.
2.3.3 Study on Asp26 in Thioredoxin
The free energy difference between protonated and deprotonated Asp26 is shown
in Figure 2-3C and D. By analogy with the model compound plots, the cumulative
average as a function of time is reported. The cumulative average was clearly not
converged during the TI simulation, and neither was the difference between forward
and backward ∆G(proteinAH → proteinA−). According to Table 2-1, after 4 ns of TI
simulation, the difference between forward and backward free energy was 1.4 Kcal/mol,
while the uncertainty of the forward and backward free energy differences was 0.61
and 0.39 Kcal/mol, respectively. Data not presented here show that TI requires roughly
43
40 ns of dynamics before converging to results comparable with FEP/REFP. It is worth
noting that this comparison is slightly unfair to TI and deserves further explanation.
First, we used eight intermediate states for TI versus 16 for FEP/REFEP. This setup,
when executed within Amber, uses the same CPU time since the TI implementation
is done with dual-topology methods. In fact, reusing the ensemble generated with
the FEP Hamiltonians and computing TI values on that ensemble produces very
rapidly-converging results.
For regular FEP free energy calculations, the cumulative averages stabilized
after roughly 2.2 ns of simulation, while the cumulative averages for the REFEP
simulation stabilized much more rapidly (shown in Figure 2-3C). Furthermore, Figure
2-3D illustrates that the difference between forward and backward ∆G(proteinAH →
proteinA−) in the REFEP reached a value very close to zero (-0.05 Kcal/mol) very
quickly. As described previously, the final value of ∆G(proteinAH → proteinA−) was
calculated as the average of forward and backward free energy differences. Although
the final free energy differences computed from 4 ns of simulation were the same for
REFEP and regular FEP, the calculations converged much faster in REFEP than in FEP
simulation. Since the HREMD and FEP calculations only differed in whether structures
were allowed to be exchanged or not, the improvement in alchemical free energy
convergence resulted from employing enhanced conformational sampling technique is
significant. Data not presented here show that the histograms of P1(∆U) exp(−β(∆U))
for the calculation of the free energy difference between replicas 1 and 2 for different
sampling times are slightly different for FEP and REFEP. The REFEP distributions
converge faster with time and sample the left side of the distribution better. This helps
rationalize the faster convergence of our technique.
2.3.4 pKa Prediction for Asp26 in Thioredoxin
The pKa value of Asp26 in thioredoxin can be computed from Equation 2–10.
The final value of ∆G(proteinAH → proteinA−) from the REFEP simulation was
44
0 1000 2000 3000 40006.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0P
redi
cted
pK
a V
alue
Time (ps)
REFEP, forward REFEP, backward FEP, forward FEP, backward
Figure 2-4. Predicted pKa value of Asp26 in thioredoxin as a function of time. The(∆G(AH→ A−)) values utilized in Equation 2–10 were -59.68 and -59.78Kcal/mol for REFEP and FEP, respectively. The experimental value is 7.5.
-54.3 Kcal/mol, with a predicted pKa value of 7.9, which is only 0.4 pKa units above
the experimental value. The predicted pKa value with respect to time from REFEP
simulations was plotted in Figure 2-4 in order to demonstrate the convergence of
the pKa prediction. Figure 2-4 shows that REFEP simulations not only yielded an
accurate predicted pKa value but also achieved convergence very fast. The regular FEP
simulation predicted a pKa value of 8.2, which is 0.7 pKa units above the experimental
value. The convergence in the regular FEP simulation was also worse than that in the
REFEP simulation.
45
2.4 Concluding Remarks
Conformational sampling is crucial in free energy calculations. In the case of
alchemical free energy calculations, HREMD is a useful and popular method to
enhance the accuracy and convergence of free energy simulations. In this chapter,
we have demonstrated that REFEP not only improves conformational sampling in
free energy calculations but also yields a free energy difference directly via the FEP
algorithm. The implementation of REFEP is trival, once a HREMD code is in place.
The REFEP alchemical free energy calculation was tested on predicting the pKa value
of Asp26 in thioredoxin and compared with TI and regular FEP simulations. Free
energy differences from the REFEP simulation converged faster than those from TI
and regular FEP simulations. The final predicted pKa value from the REFEP simulation
was very accurate, only 0.4 pKa unit above the experimental value. Utilizing the REFEP
algorithm significantly improves conformational sampling, and this in turn improves the
convergence of alchemical free energy simulations.
46
CHAPTER 3LARGE PKa SHIFTS FOR BURIED PROTONABLE RESIDUES. RATIONALIZING THE
CASE OF GLUTAMATE 66 IN STAPHYLOCOCCAL NUCLEASE
3.1 Literature Review
The pKa of an ionizable species is related to the equilibrium constant between
its protonated and deprotonated states. Because the electrostatic potential and the
protonation equilibrium are coupled, protein electrostatics can be probed indirectly by
pKa measurements. Ionizable groups are usually solvent-exposed and situated near
the protein surface, where the pKa shift (as measured versus the same residue by itself,
in the same solution, usually called a model compound) of these groups is relatively
small and ascribed to the coulombic interactions with other ionizable residues and salt
bridges. On rare occasions, proteins have buried ionizable residues that behave very
differently from those located near the surface. These exceptional cases can be very
useful to understand protein electrostatics.
Buried titratable groups can play critical roles in a variety of situations[79–82]. The
charged forms of ionizable groups are more favorable in strongly polar environments,
while the neutral species of ionizable groups are dominant in hydrophobic environments.
This preference for the neutral species in the hydrophobic interior of proteins causes
a large shift in the pKa of buried titratable groups , acidic groups have pKa values
shifted significantly higher and basic groups have pKa values shifted significantly lower
[83, 84]. Measuring and rationalizing the pKa of internal ionizable groups is crucial to
understanding biochemical processes such as charge transfer [82, 85–87], molecular
recognition [88], and ion transport [89, 90]. Electrostatic interactions between titratable
sidechains govern all pH-dependent properties of proteins [91].
Recent studies show that the hydrophobic interior of some proteins can tolerate
charged residues without any significant structural adaptation. Garcia Moreno
et al. have studied the stability and pKa shift of E. coli’s staphylococcal nuclease
(SNase) mutants using fluorescence, circular dichroism spectroscopy (CD), and other
47
experimental methods [83, 84, 92–98]. In one of those studies, a valine-to-glutamate
mutation at position 66 (V66E) has been introduced to ∆ + PHS, a hyper-stable mutant
of SNase. This substitution results in a highly shifted pKa value for glutamate (GLU) 66
compared to that of the GLU model compound. Different experimental methods have
reported this shifted value as 9.0 ± 0.2 (i.e., ∆pKa ≈ 4.6) [92, 94, 99, 100], making it
one of the largest pKa shifts ever reported for a titratable residue. On the other hand,
continuum solvent computational methods (e.g., the Generalized Born model) predict a
smaller pKa shift for GLU66 [97, 99].
According to the Born formulation [101, 102], this shift corresponds to the transfer
of a charged group from water to a medium of dielectric constant of 10-12. This value,
required for improved agreement between continuum solvent methods and experimental
results in this case, is much larger than the usual value of 2 to 4[101, 103]. There is no
obvious physical rationale behind this high value of dielectric constant [91, 97, 99, 100].
Therefore different reasons have been proposed to rationalize this discrepancy. Some
of the most significant ones are changes in the state of ionization of other residues,
interaction with buried water, protein relaxation[83, 84], and interactions with other
ionizable residues[91, 97, 99, 104]. Even after accounting for these effects, a dielectric
constant around 6 is still necessary to reproduce experimental data[91, 97, 99, 105].
In this chapter we show that to understand these systems properly, one needs to
explicitly treat the conformational and protonation equilibria as coupled, and there is no
single structure that can be seen as responsible for the pKa switch.
3.2 Simulation Details
We built the V66E mutant of the hyperstable form of staphylococcal nuclease
(SNase) known as ∆ + PHS (pdb code 3BDC17). As before, all simulations were
performed using the Amber FF99SB force field [76] as implemented in the Amber
10 molecular simulation package [75]. The SHAKE algorithm [77] was used to keep
bonds involving hydrogen atoms at their equilibrium length. Newton’s equations were
48
integrated with a timestep of 2 fs. We minimized the system using 500 steps of the
steepest descent algorithm, followed by 500 steps of the conjugate gradient algorithm.
Next, we heated the system to 300 K in three steps, followed by a final equilibration in
the canonical ensemble. All calculations were performed using the Langevin Thermostat
[106] and distinct random seeds (ig=-1) in order to generate independent simulations
[107]. The generalized Born implicit solvent model [102, 108] was applied to model the
water environment in all calculations. The cutoff used for non-bonded interactions
and calculating Born radii was 999 A. We used the recently developed Replica
Exchange Free Energy Perturbation method (REFEP)[22] (Details in chapter 2) to
calculate the free energy difference between the protonated and deprotonated states
in each conformational state (i.e., either buried or exposed GLU66) and in a mix of
conformational states.
Three sets of REFEP simulations were performed on this system: (1) conformationally
unrestrained, (2) a 0.4 kcalmol−1A−2 harmonic restraint on all heavy atoms in the
conformation with GLU66 buried and (3) the same restraints in the conformation where
GLU is exposed to the solvent. In all three sets of simulations, 16 replicas were used to
perturb the protonated GLU66 (λ = 0) to the deprotonated state (λ = 0). The length of
the simulation was 5 ns for simulation(1) and 4 ns for simulations (2) and (3). Also a 5 ns
REFEP simulation was performed on a Glu reference compound (i.e., Ace−Glu− Nme)
using 16 replicas to perturb the protonation state of Glu to establish the deprotonation
free energy of a free Glutamic acid in solution.
Hereafter, we will refer to the pKa of GLU66 as the pKa of GLU66 in V66E ∆ + PHS
and to the pKa shift as the difference between this value and that of the reference
compound in water (4.4).
In this work, we used the CDPro software package [109] to calculate the Circular
Dichroism (CD) spectra as the protonation of the system is modified. CDPro computes
49
the CD spectra as function of conformation, using parameterized dipole-dipole
interactions[110].
3.3 Discussion
According to experimental methods such as X-ray, TRP Florescence, and CD
Spectroscopy, ionizing the GLU66 side chain in V66E ∆ + PHS triggers the unwinding of
one turn of an α−helix, exposing the previously buried carboxylic group of GLU66 to the
bulk solvent, shown in Figure 3-1.
Figure 3-1. one turn of an α−helix exposes the side chain of GLU66.
As explained in the introduction, there is inconsistency between pKa shift calculated
by experimental methods and continuum solvent computational methods. In this part we
tried to rationalize this inconsistency.
We used REFEP to calculate the free energy difference between the two-protonation
states for both the GLU reference compound and GLU66 in V66E ∆ + PHS and turned
that data into a pKa shift due to the protonation. The REFEP data is shown in Figure
50
3-2. Comparing the final free energies in both directions (Figure 3-2) shows that both
53 54 55 56 57 58 59 60 61 62
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
fre
e e
ne
rgy
(kc
al/
mo
l)
simulation time (ns)
A) Final backward free energy
V66E ∆+PHS, UnrestrainedReference compound
-62
-61
-60
-59
-58
-57
-56
-55
-54
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
fre
e e
ne
rgy
(kc
al/
mo
l)
simulation time (ns)
B) Final forward free energy
V66E ∆+PHS; UnrestrainedReference compound
Figure 3-2. Free energy convergence in REFEP for unrestrained V66E ∆ + PHS (red)and Glu model compound (green). The cumulative free energy differencesare between the highest (fully deprotonated) and the lowest (fullyprotonated) replica For A) Backward and B) Forward free energy differences.
REFEP calculations of unrestrained V66E ∆ + PHS and Glu compound are converged.
We found that the GLU66 pKa in unrestrained is shifted about 3.2 pKa units higher
relative to the model compound, roughly 1.5 pKa units less than the experimental
value[92, 104]. Analyzing this data, it becomes obvious that changing the protonation
coordinate during REFEP, also induced a local conformational change around the loop
containing residue 66 (See Figure 3-1). Accordingly, translating the computed free
energy difference for the protein into a pKa is not correct.
51
At this point, we present the most important result of this chapter: so far many
macroscopic and microscopic models have been proposed for estimation and calculation
of pKa in proteins [22, 111–115]. The microscopic approach in this chapter is grounded
on calculation of protonation/deprotonation energies at a given conformation. Based
on that, the proper treatment of these systems requires explicit inclusion of coupled
protonation and conformational equilibria[116]. Basically, at low pH, GLU66 is both
buried and protonated, while at high pH it is solvent exposed and deprotonated. To
understand this system we introduce a conformation-protonation equilibrium model,
shown in Figure 3-3, to calculate the pKa of GLU66 in the SNase mutant. Using this
model, we distinguished between conformational states of the GLU66 sidechain (i.e.,
e=exposed/b=buried) and its protonation states (i.e., P=protonated/D=deprotonated).
Figure 3-3. Thermodynamic cycle for conformation protonation model of GLU66. Db andPb stand for deprotonated-buried and protonated-buried, respectively; andDe and Pe stand for deprotonated-exposed and protonated-exposed,respectively.
The horizontal line at the top corresponds to the protonation change at the exposed
conformation, the horizontal line at the bottom to the protonation change at the buried
conformation, while the two vertical lines are the free energies required to change
52
conformations at fixed protonation states. This separation allows us to study the
conformational and protonation changes separately and coupled, to better understand
the results.
To calculate the pKas in the buried and exposed cases individually, we performed
two sets of restrained REFEP simulations. In the buried case, the GLU66 side chain was
restrained to the interior of the protein, and in the exposed case, it was restrained to stay
exposed to the solvent by applying a harmonic restraint on all heavy atoms.
Restrained REFEP calculations (shown in Figure 3-4) compute a pKa of 10.6 for the
buried situation and of 4.6 for the exposed GLU66 sidechain, respectively.
52
53
54
55
56
57
58
59
60
61
62
0 0.5 1 1.5 2 2.5 3 3.5 4
free e
nerg
y(k
cal/m
ol)
simulation time (ns)
A) Final backward free energy
GLU66 side chain restrained-buriedGLU66 side chain restrained-exposed
-62
-61
-60
-59
-58
-57
-56
-55
-54
-53
-52
0 0.5 1 1.5 2 2.5 3 3.5 4
free e
nerg
y(k
cal/m
ol)
simulation time (ns)
B) Final forward free energy
GLU66 side chain restrained-buriedGLU66 side chain restrained-exposed
Figure 3-4. Cumulative Free energy VS Simulation time for Glu-66 restrained inside(Red) and restrained outside (Green).
53
The thermodynamic cycle in Figure 3-3 is written in such a way that knowledge
of the two microscopic pKas and one of the equilibrium constant (or free energy) for
the conformational change, fully fixed the value of the second equilibrium constant.
Using mass balance equations, we have a system that has a single free parameter,
corresponding to one of the two vertical lines in Figure 3-3.
Since most experimental probes are sensitive to conformations, we will compute an
equilibrium constant that depends on either buried of exposed GLU66, disregarding
protonation states. The equilibrium constant between the exposed and buried
conformations of the GLU66 side chain can be written as:
Ke/b =[De] + [Pe]
[Db] + [Pb]=
[De]
[Db]
1 + 10(pKa,e−pH)
1 + 10(pKa,b−pH)= Ke/b,D
1 + 10(pKa,e−pH)
1 + 10(pKa,b−pH)(3–1)
Here, Ke/b,D is the Ke/b at high pH (i.e., when GLU66 is deprotonated).
Using the cycle presented in Figure 3-3, analytical expressions of the concentration
of each species as a function of the equilibrium constants and pH can be derived.
[De] =(
Ke/b
1+Ke/b
)(1
1+10(pKa,e−pH)
)[Pe] =
(Ke/b
1+Ke/b
)(10(pKa,e−pH)
1+10(pKa,e−pH)
)[Db] =
(1
1+Ke/b
)(1
1+10(pKa,b−pH)
)[Pb] =
(1
1+Ke/b
)(10
(pKa,b−pH)
1+10(pKa,b−pH)
). (3–2)
These expressions can be used to calculate the concentration of these species for
different values of Ke/b,D (Figure 3-5). For high values of Ke/b,D, the exposed-deprotonated
specie (De) is dominant at High pH value. The overall pKa can be defined in terms of a
protonation equilibrium constant, i.e.,
KD/P =[De] + [Db]
[Pe] + [Pb](3–3)
However, most experimental methods are sensitive to conformations and do not directly
report on the protonation state of a specific residue. Based on this, we can define a
54
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
2 4 6 8 10 12
co
ncen
trati
on
pH
pKa;e = 4.6
pKa;b = 10.6
Ke/b,D;1 = 2
Ke/b,D;2 = 1e+08
[De]1[Pe]1[Db]1[Pb]1[De]2[Pe]2[Db]2[Pb]2
Figure 3-5. Concentration of species vs. pH for two different value of Ke/b,D i.e.,Ke/b,D;1 = 2 and Ke/b,D;2 = 108.
Conformation-Sensitive Signal (CSS) for the case of GLU66 in V66E ∆ + PHS as:
CSS = [De] + [Pe]− [Db]− [Pb] (3–4)
At a very high pH, GLU66 is predominantly deprotonated and only species where
GLU66 is deprotonated will contribute significantly to the observed CSS. On the other
hand, at a very low pH, only the protonated species will be significant, i.e., CSSpHhigh= [De]− [Db]
CSSpHlow= [Pe]− [Pb]
. (3–5)
55
Figure 3-6 shows the CSS, as defined in Equation 3–4, as a function of pH. The pH
at which the inflection point is observed corresponds to the experimentally-measured,
apparent pKa. If the GLU66 is fully exposed at high pH and fully buried at low pH, then
the observed inflection point will correspond to a zero value of the CSS. However, if
there is a mixture of buried and exposed states, then the inflection point will be shifted
to a negative signal value. Based on Equation 3–1 and Equation 3–2, it is possible to
calculate the experimental inflection point as a function of pKa,e, pKa,b, and Ke/b,D.
Figure 3-6. CSS, as defined in Equation 3–4, as a function of pH for Ke/b,D=10; theorange arrows point to the inflection point ( i.e., half way between the highpH and the low pH values).
Figure 3-7 shows the apparent pKa plotted as a function of Ke/b,D. Figure 3-7
demonstrates that the pKa, which is measured using experimental methods that are
56
sensitive to conformations, depends on the ratio of two distinct conformational states at
high pH. This model predicts that the apparent pKa ranges between 4.6 (Ke/b,D ≫1,
fully solvent exposed) and 10.6 (fully buried, Ke/b,D ≪1). Ke/b,D modulates the apparent
value of pKa. As Ke/b,D increases, the apparent pKa decreases. As Ke/b,D approaches
4
5
6
7
8
9
10
11
0.0001 0.01 1 100 10000 1e+06 1e+08
ap
pare
nt
pK
a
Ke/b,D
Figure 3-7. Apparent pKa vs. Ke/b,D.
infinity, the apparent pKa tends to 4.6.
We estimated Ke/b,D based on Root Mean Square Deviation (RMSD) distributions
of the GLU66 side chain and part of its belonged α−helix (residue62-69), for all the
replicas in the unrestrained REFEP simulations (Figure 3-8). In Figure 3-8, for λ=0
(GLU66 is protonated) we can identify two peaks at 0.4 and 1.1 A, with the former being
the highest. As λ increases, a new population around 1.6 A will appear and the others
will disappear. At λ=1.0 the peak at 0.4 A has disappeared, but there is still a small peak
at 1.1 A. Because of overlap between protonated and deprotonated RMSD distributions
57
λ= 0.0
RMSD(A⋅ )
Fre
quen
cy
0.0 0.5 1.0 1.5 2.0 2.5
010
025
0λ= 0.067
0.0 0.5 1.0 1.5 2.0 2.5
010
025
0
λ= 0.133
0.0 0.5 1.0 1.5 2.0 2.5
010
025
0
λ= 0.2
0.0 0.5 1.0 1.5 2.0 2.5
010
025
0
λ= 0.267
0.0 0.5 1.0 1.5 2.0 2.5
010
025
0
λ= 0.333
0.0 0.5 1.0 1.5 2.0 2.5
010
025
0
λ= 0.4
0.0 0.5 1.0 1.5 2.0 2.5
010
025
0
λ= 0.467
0.0 0.5 1.0 1.5 2.0 2.5
010
025
0
λ= 0.533
0.0 0.5 1.0 1.5 2.0 2.5
010
025
0
λ= 0.6
0.0 0.5 1.0 1.5 2.0 2.5
010
025
0
λ= 0.667
0.0 0.5 1.0 1.5 2.0 2.50
100
250
λ= 0.733
0.0 0.5 1.0 1.5 2.0 2.5
010
025
0
λ= 0.8
0.0 0.5 1.0 1.5 2.0 2.5
010
025
0
λ= 0.867
0.0 0.5 1.0 1.5 2.0 2.5
010
025
0
λ= 0.933
0.0 0.5 1.0 1.5 2.0 2.5
010
025
0
λ= 1.0
0.0 0.5 1.0 1.5 2.0 2.5
010
025
0
Figure 3-8. Carbon α RMSD distributions of residues 62 to 69 and the side chain ofGLU66 in unrestrained REFEP simulations for all sixteen replicas withrespect to the average structure.
(λ=0 and 1.0), we can conclude that both protonation states are ensembles of both
buried and exposed configurations. This shows that the changes in protonation states
are not fully synchronized with the changes in conformation states.
For more rigorous investigation of the decoupling, we calculated the CD spectra and
average secondary structure for all λ values in the unrestrained simulations. Average
secondary structure shows that as λ increases, the α−helix content decreases and
310 helix content increases (Figure 3-9). Figure 3-10 shows the ellipticity at 222 nm,
58
0
10
20
30
40
50
60
70
80
0 0.2 0.4 0.6 0.8 1
perc
en
tag
e o
f th
e s
eco
nd
ary
str
uctu
re
λ
3-10-Helixα helix
turn
Figure 3-9. Average secondary structure for residue 57-69 vs. reaction coordinate.
which is the wavelength at which the finger print of α−helix appears in CD, for the whole
protein and the region containing Glu66 (i.e., residues 57-69). The α−helix content
decreases as the system shifts toward the deprotonated Glu66. The trend observed for
the whole protein and the local region is the same and is consistent with the fact that the
deprotonation of GLU66 locally denaturates part of α−helix content of that region. The
results of Figure 3-10 can be rationalized by looking at Figure 3-8. In the latter, RMSD
histograms of λ ∼ 0.2-0.4 show the maximum frequency on the leftmost part of the
RMSD axis (less than 1.3 A). These are the λs that have the maximum absolute value
of ellipticity in Figure 3-10. Estimating RMSD ∼ 1.0-1.3 A as the boundary between
an exposed and a fully buried GLU66 side chain, when λ=1.0 less than 1.8-6.7 % of
59
-7.15E+04
-7.10E+04
-7.05E+04
-7.00E+04
-6.95E+04
-6.90E+04
-6.85E+04
-6.80E+04
-6.75E+04
0 0.2 0.4 0.6 0.8 1 -2.05E+04
-2.00E+04
-1.95E+04
-1.90E+04
-1.85E+04
-1.80E+04
-1.75E+04
-1.70E+04
-1.65E+04
-1.60E+04ellip
ticit
y
λ
whole proteinresidues 57-69
Figure 3-10. Ellipticity at 222 nm vs. reaction coordinate ( λ=0, i.e., GLU66 protonatedand λ=1, i.e., GLU66 deprotonated ) for whole protein(red, left y axis) andfor the region containing Glu66 (green, right y axis)
the population is in a fully buried conformation. Based on these results, Ke/b,D can be
estimated to be ∼ 15-57, which yields an apparent pKa of 8.7-9.2 (Figure 3-7). This
agrees well with the results of structure sensitive experimental methods, which estimate
the pKa of the GLU66 side chain to be 9.0-9.1 [92].
3.4 Concluding Remarks
There are many challenges in determining pKa shifts of buried ionizable groups.
The mutation of position 66 in staphylococcal nuclease is an interesting example for two
reasons. First, according to experimental methods the shift in pKa of ionizable group
mutants range from 3.5 -5.0, depending on the ionizable group and their relative normal
60
pKa. Second, there is an inconsistency between pKa shifts calculated by implicit solvent
methods and those of experimental methods. To solve this challenging problem, we
used REFEP to calculate the pKa shift of staphylococcal nuclease mutant (V66E∆ +
PHS).
Not only the conformational response of a protein to a change in protonation state
of an ionizable group may be beyond the sensitivity of experimental methods, but also
conformational changes may not be fully correlated to the changes in protonation state.
The fact that protonation changes are not necessarily accompanied with configurational
changes shows the robust threshold of a protein against local electrostatics variations.
Based on this fact, the inconsistency between implicit solvent simulations and
conformational sensitive experimental methods has been explained. A four-state
thermodynamic cycle has been introduced to decipher this inconsistency. Through the
thermodynamic cycle, four species have been defined, which correspond to all possible
combinations of conformational-protonation of the GLU66 sidechain. We have shown
that the inflection point of experimental titration curves is dependent on the ratio of
conformational states at high pH; by calculating the ratio of conformational states at high
pH, the experimental inflection point that we have determined is in good agreement with
the one obtained from the experimental results.
61
CHAPTER 4PH-REPLICA EXCHANGE MOLECULAR DYNAMICS IN PROTEINS USING A
DISCRETE PROTONATION METHOD
4.1 Literature Review
Solution pH plays a very important role in protein function, dynamics and structure.
Many important biological phenomena function only at a certain range of pHs, including
protein folding/misfolding[117–119], enzyme catalysis[120–122] and ligand-substrate
docking[123, 124]. In those cases, a pH change leads to a change in the ratio of
protonation states of different titratable residues, which is usually coupled to a change
in the conformation and dynamics of the protein itself. The pKa of a titratable residue
is the pH value at which the ratio of deprotonated to protonated concentrations of that
residue is equal to one[72, 112, 125, 126]. The pKa of an ionizable residue in a protein
is highly dependent on its electrostatic environment, which is coupled to the protonation
states of other titratable groups and the conformation state of the protein. Most current
simulations of pH dependent properties do not, in fact, change protonation states
during molecular dynamics, but rather pick a certain set of protonation states using
educated guesses, or guided by simplified algorithms, and then keep it constant for the
remainder of the simulation. These constant protonation MD methods suffer from two
big disadvantages[127]. First, at a pH near the pKa of any of the titratable residues, any
possible choice of constant protonation would clearly be wrong. Second, the choice
of protonation state is coupled for neighboring residues, which itself couples to the
conformational space of the system.
Like concentration and temperature, the solution pH is a very useful external and
controllable variable in experimental methods. Hence, the importance of constant
Reprinted with permission from Sabri Dashti, D.; Meng, Y.; Roitberg, A. E. J. Phys.Chem. B 2012, 116, 8805–8811.
62
pH MD (CpH MD) has been recognized. In the last two decades, many constant pH
MD methods have been developed[128, 129]. The goal of CpH MD is to describe
correctly the protonation equilibrium coupled by conformation equilibrium at a certain
pH. The majority of CpH MD methods can be divided into continuous[128–136]
and discrete[127, 137–146] protonation state methods. Continuous protonation
methods use a continuous protonation parameter to perturb ionizable residue between
protonated and deprotonated states. In 1994, Mertz and Pettitt[135] developed a
grand canonical method for simulating a simple chemical reaction. They applied the
method for exchanging a proton between water molecules and an ionizable side chain.
In 1997, Baptista et al.[128] introduced a continuous constant pH method in implicit
solvent based on a mean-field approximation. In 2001, Borjesson et al. [129] used
a weakly coupled proton bath to continuously adjust the protonation fraction of each
titratable group towards equilibrium. More recently, the Brooks group has developed
the continuous protonation state method further[130–134, 147]. In the case of highly
coupled titration groups, for which cooperativity effects are non-negligible, this model
leads to inappropriate estimation of physical variables. To alleviate this problem, Lee et
al.[147] added a biasing potential, centered at λ equal 0.5, to help drive the protonation
coordinate value to fully protonated/deprotonated states, and away from the mid-λ
unphysical states.
In contrast to the continuous protonation state methods, discrete protonation
models define the protonation state of the ionizable group as either zero or one during
the simulation, corresponding to protonated and deprotonated states only. These
models use a hybrid MD-MC scheme; the MD is used to sample conformational space
for a number of steps, after which a Metropolis MC[14] attempt is done for changing
the protonation state/states. A new set of MD steps is then done with the protonation
state chosen by the MC step, and the process is repeated. Many versions of the
constant pH/discrete protonation MD method have been developed[115, 127, 138–
63
143, 145, 146, 148]. The Baptista group[137–140, 148] used explicit solvent for the
propagation of coordinates and the Poisson-Boltzmann (PB) method for calculating
the energy in the MC section of the algorithm. Walczak et al. [146] employed Langevin
Dynamics for MD and the PB method for both of the MC and MD steps. Burgi et al.[141]
applied Thermodynamic Integration (TI) to calculate the transition energy between
the protonated and deprotonated species in explicit solvent MD. The drawback for this
approach is the large computational cost of TI calculations, which limits the amount of
sampling in protonation space. In 2004, Mongan et al. developed a discrete protonation
MD method by using the GB implicit solvent method to both the MD (structure) and MC
(protonation state) sampling sections[127]. A more detailed explanation of the method
can be found in the methods section. This method is implemented in the AMBER
Molecular Dynamics Package[75].
It has become clear in recent years that accurate modeling of protonation space
also requires enhanced sampling of conformational space[149–151]. Accurate
sampling of conformational space of proteins remains a challenging area [2, 3, 18,
152, 153, 155–157]. Many theoretical methods have been proposed to overcome
the free energy barriers in conformational space (see chapter 1). To account for the
coupling of protonation and conformational sampling, Wallace et al. [158] recently
have combined the continuous protonation Constant pH MD with the REMD method
(REX-CPHMD)[131]. They applied it to the problems of pKa prediction in protein
folding and pH dependent conformation, among others[158, 159]. Recently, Meng and
Roitberg[160] utilized a hybrid method by combining the Temperature REMD (TREMD)
and discrete protonation Constant pH MD. In this chapter, we introduce a method for
pH-Exchange MD by combining the discrete protonation Constant pH MD(proposed
by Mongan et al.[127]) and Hamiltonian REMD (HREMD) [22]. We tested our method
by applying it to five model dipeptides, to an uncapped pentapeptide with sequence
+H3N−Ala−Asp− Phe−Asp−Ala− COO− (ADFDA), and to an heptapeptide derived
64
from the ovomucoid third domain OMTKY3 protein. Because the two ends of ADFDA
are not capped, its two Asp residues have slightly different electrostatic environments
and their pKas deviate in different directions from the pKa of unperturbed Asp. We will
show that the pHREMD MD improves the sampling efficiency in both protonation and
conformation spaces.
In the rest of this chapter, Constant pH MD (CpH) refers to Mongan’s CpH MD
approach unless mentioned otherwise.
4.2 Theoretical Method and Simulation Details
4.2.1 Constant pH Molecular Dynamics and pH-Replica Exchange MolecularDynamics (pH-REMD)
The goal of CpH MD is to sample the equilibrium between protonated and
deprotonated state of titratable sites at a given pH. The free energy difference between
protonated and deprotonated states determines the ratio of their concentrations. This
free energy difference can not be calculated by Molecular Mechanics (MM), because
the change in protonation state involves a bond breaking/forming phenomenon, which
requires a series of highly accurate quantum calculations. To address this issue, a
method[115, 127, 161], which uses a pre-calculated pKa of reference compounds, has
been developed. Reference/model compounds, in AMBER terminology, are represented
by a capped dipeptide for each titratable residue (i.e., ACE-titratable residue-NME). The
free energy for the protonation change to be used in the MC criteria is described by
Mongan et al. [161] as:
∆Gprotein = ∆Gprotein,MM + kBT(pH− pKa,ref) ln 10−∆Gref,MM. (4–1)
Here, as before, T is the temperature, and kB is the Boltzmann constant. ∆Gprotein,MM is
the molecular mechanics part of the free energy of the titratable site in the protein,
and ∆Gref,MM is the pre-computed deprotonation free energy for the reference
compound, described above. Using Equation 4–1, there is no need to calculating
65
the QM contribution to the free energy in the protein. This method is implemented in
the AMBER MD suite[75], using the GB implicit solvent model. Every few MD steps
a Metropolis MC[14] attempt is done to change the protonation state of the titratable
residue and ∆Gprotein is used to make a decision about accepting or rejecting the
proposed MC move. In other words, the MC moves sample protonation space, and the
MD steps sample configuration space. During the MD steps, the protonation state is
kept constant.
4.2.2 Titration Curve
The pKa of a titratable residue is related to the pH environment through the
Henderson-Hasselbalch (HH) equation,
pKa = pH− n log
([A−]
[HA]
), (4–2)
with [A−] and [HA] being the deprotonated and protonated concentrations respectively; n
is the Hill coefficient, which should approach 1 for non-interacting ionizable residues, but
deviate from one in the case of interacting titratable residues because of cooperativity[162].
Because of the ergodicity assumption underlying MD, the ratio of the time that a
titratable residue spend at deprotonated state to the time spend in protonated state can
be considered as a ratio of concentration of deprotonated state to that of protonated
state.
On the other hand, pH-REMD is a combination of CpH[161] and HREMD[22]. In
contrast to temperature replica exchange[2], in which each replica runs at a different
temperature, in HREMD, each replica runs in a distinct Hamiltonian but at the same
temperature. In pH-REMD each replica runs a constant pH MD at a unique pH, and
periodically an exchange of conformation between two adjacent replicas is attempted
(section 4.2.3).
66
We note that a recent publication[23] presented a pH replica exchange method
using a different exchange criterion, that does not require recomputing energies for
different replicas. In the future, we will compare the two formulations.
4.2.3 pH-REMD
In the HREMD algorithm, each replica runs a distinct Hamiltonian. In pH-REMD
each replica runs a CpH MD simulation at a unique pH, with a swap of conformations
between two adjacent replicas is attempted regularly. Using the detailed balance
condition, we can write an equilibrium proposition for the ensemble before and after an
exchange is attempted,
w(X→ X)℘(X) = w(X→ X)℘(X), (4–3)
where ℘(X) is the probability of being at state X and w(X → X) is the probability of
transition from state X to that of X.
At the exchange moment, if we only swap the conformations between two adjacent
replicas and keep the protonation state unchanged, then the generalized states of X and
X can be written as
X =
n1 ... ni nj ... nM
q1 ... qi qj ... qM
, X =
n1 ... nj ni ... nM
q1 ... qi qj ... qM
, (4–4)
Here q represents a conformation and n represents protonation states. The ith column is
related to the ith replica.
Because all M replicas are independent, the probability of the system being in the
generalized state X can be written as ℘(X) =∏M
i=1 Pi(qi, ni), where Pi(qi, ni) is the
probability of replica i at conformation qi and protonation state of ni. Substituting those
probabilities in Equation 4–3 will result
w(X→ X
)w(X→ X
) =P(qj, ni)P(qi, nj)
P(qi, ni)P(qj, nj)= exp(−β∆), (4–5)
67
In the canonical ensemble regime (NVT), and using the Boltzmann distribution as a
limiting ensemble, this can be written as
w(X→ X
)w(X→ X
) =exp(−βH(qi, nj)) exp(−βH(qj, ni))
exp(−βH(qi, ni)) exp(−βH(qj, nj))= exp(−∆), (4–6)
where β = 1kBT
and ∆ = [H(qi, nj +H(qj, ni)− H(qi, ni)− H(qj, nj)].
This setup can be realized by setting the exchange probability according to the
Metropolis MC criteria as
w(X→ X
)= w ((qi, ni); (qj, nj)→ (qj, ni); (qi, nj)) = min (1, exp(−∆)) . (4–7)
For computing ∆ only the potential energies are required. Since the two exchanging
replicas are running at the same temperature and have the same number and mass of
particles, their kinetic energy terms will cancel.
4.3 Simulation Details
To validate and test the method presented here, we chose and ran simulations
on three categories of systems. First, we studied the capped reference compounds
(described in the methods section), consisting of the ACE − titratable residue − NME.
Simulation times were 3 ns for both CpH MD and pH-REMD (for each replica) methods.
We used eight pH-replicas for all the model compounds in pH-REMD. We also tested
our simulation method on a terminally charged pentapeptide model, +H3N − Ala-
− Asp − Phe − Asp − Ala − COO− (ADFDA). Because the two ends of ADFDA are
oppositely charged, the two Asp experience different electrostatic environments and
have different pKas. The simulation times of ADFDA were 90 ns for CpH and 10 ns
for each replica in pH-REMD. Eight replicas, at pH=2.5-6.0 with increment of 0.5 were
used in pH-REMD method. The third system was a heptapeptide derived from OMTKY3
(ACE − Ser − Asp − Asn − Lys − Thr − Tyr − Gly − NME). We simulated 100 ns for
CpH and 10 ns for pH-REMD. Twelve replicas, at pH=2-13 with increment of 1 pH unit,
were used in pH-REMD method. All calculations in this chapter were done using the
68
AMBER 10 molecular simulation suite. The AMBER ff99SB force field[76] and OBC
Generalized Born implicit solvent[77] (igb=2) were used in all simulations. The Shake
algorithm [101] was applied to constrain the bonds between heavy and hydrogen atoms,
which allowed the use 2 fs MD steps. A cutoff of 30 A for non-bonded interactions was
chosen in all calculations. In all pH-REMD simulations, except the heptapeptide derived
from OMTKY3, replicas attempted a pH exchange every 1000 MD steps. In the case of
the heptapeptide, replicas attempted a pH exchange every 500 MD step to accelerate
sampling[163]. The exchange acceptance ratios between replicas for all system were
between 0.2 and 1.0. For calculating the error bars and the uncertainty of pKas, every
4000 MC steps (in protonation space), the deprotonation fraction has been calculated.
4.4 Results and Discussion
4.4.1 Titratable Model Compounds
When we apply pH-REMD to the model compounds that were initially used
to parametrize the method, it is not surprising to find that it produces the correct
pKas. It is, however, an important calculation to perform to check the method and to
gauge its efficiency versus constant pH runs. The results are shown in Table 4-1,
where all pKa values have been calculated by a fit to the linearized version of the
Henderson-Hasselbalch Equation.
Table 4-1. pKas of the reference compounds computed by different methodsASP GLU HIS TYR LYS
Experimental value 4.0 4.4 6.3 9.6 10.5CpH MD 3.9±0.1 4.4 ±0.1 6.3±0.1 9.6±0.03 10.4±0.03pH-REMD 3.9±0.1 4.4 ±0.1 6.4±0.1 9.7±0.02 10.4 ±0.04
In Figure 4-1, the Hill plot for capped Lysine is shown for both pH-REMD and CpH
MD methods, with agreement between them over a large pH range.
4.4.2 ADFDA Model Compounds
We now apply the method to the model peptide, +H3N − Ala − Asp − Phe − Asp-
− Ala− COO− (ADFDA), that has charged ends. This system has an intrinsic but subtle
69
-2
-1.5
-1
-0.5
0
0.5
1
1.5
9 9.5 10 10.5 11 11.5
log 1
0 ([
A- ]/
[AH
])
pH
pkaCpH= 10.4
pkapH-REMD= 10.4
HH fit CpH
HH fit pH-REMD
Figure 4-1. Comparison of Hill plots between pH-REMD and CpH methods for Lysreference model.
asymmetry in the electrostatic environment for the two Asp ionizable side chains. Asp2
is close to the NH+3 , which causes its pKa to be shifted slightly below 4.0. Asp4 is close
to the COO− terminal, which causes its pKa to be shifted above 4.0.
The plotted titration curves of Asp2 and Asp4 in Figure 4-2 have Asp2 shifted to the
left of the model compound, and Asp4 shifted to the right. There is a good agreement
between the pH-REMD and CpH curves. The predicted pKas and Hill coefficients for
both methods have been calculated by linear fitting on a Hill plot.
Table 4-2. pKa prediction and Hill coefficient of fitted from the HH equation.ASP2 ASP4
pH Hill coefficient pH Hill coefficientpH-REMD 3.8±0.1 0.89 4.5±0.1 0.88CpH MD 3.8±0.1 0.91 4.5±0.1 0.86
70
0
0.2
0.4
0.6
0.8
1
2.5 3 3.5 4 4.5 5 5.5 6
Der
oton
atio
n F
ract
ion
pH
Asp2; CpH
Asp4; CpH
Asp2; pH-REMD
Asp4; pH-REMD
Asp; reference
Figure 4-2. The titration curves of Asp side chains in ADFDA computed by both constantpH MD (blue and purple) and pH-REMD (red and green) methods; Thetitration curve of Asp for the reference compound (light blue) is also shown.
Table 4-2 shows that the results of both methods are identical. It is worth noting that
the free energy difference associated with the two different pKas is 0.96 kcal/mol, which
highlights the sensitivity of the method to very small environmental changes.
To gauge the comparative efficiency of CpH and pH-REMD, in Figure 4-3, the
cumulative average protonation fractions of Asp2 and Asp4 for both methods at pH=4.0
have been plotted. The data clearly shows that for both titratable groups, pH-REMD
converges more rapidly in protonation space than the regular CpH method.
4.4.3 Heptapeptide Derived from OMTKY3
We applied the pH-REMD method to a capped heptapeptide derived from OMTKY3
(ACE−Ser−Asp−Asn−Lys−Thr−Tyr−Gly−NME). Dlugosz and Antosiewicz[142, 143]
studied this heptapeptide and predicted the pKa of 4.24 for Asp3 using their CpH MD
71
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 100000 200000 300000 400000 500000 600000
Pro
tona
tion
Fra
ctio
n
MC Titration Steps
Asp2; pH-REMD
Asp4; pH-REMD
Asp2; CpH
Asp4; CpH
Figure 4-3. Cumulative average protonation fractions of Asps side chains in ADFDA VSMC titration steps at pH=4.0.
method. According to NMR experiments[144, 145], the pKa of Asp in this heptapeptide
is about 3.6.
Table 4-3. pKa values of the titratable residues in the heptapeptide derived fromOMKTY3.
ASP3 Lys5 Tyr7pH-REMD 3.6±0.2 10.6±0.1 10.1±0.1CpH MD 3.7±0.2 10.6±0.1 9.9±0.1
From a Hill plot, the pKa of the three titratable groups has been calculated with the
results presented in table 4-3. Our computed for values the Asp3 pKa of 3.7 (CpH) and
3.6 (pH-REMD) are in excellent agreement with the experimental value. The titration
curves are shown in Figure 4-4.
There are two more titratable groups in the heptapeptide, i.e., Lys5 and Tyr7, which
we also titrated. Figure ?? presents the titration curves of Lys5 and Tyr7, with the
72
0
0.2
0.4
0.6
0.8
1
2 3 4 5 6 7
Dep
roto
natio
n F
ract
ion
pH
Asp3; CpH Asp3; pH-REMD
Figure 4-4. The titration curves of Asp3 in the heptapeptide derived from OMTKY3.
computed pKas listed in Table 4-3. To compare the convergence speed between CpH
and pH-REMD, we studied the cumulative average protonation/deprotonation fraction
as function of MC titration steps for all ionizable residues. Figure 4-5 shows the data for
Tyr7. It is evident that pH-REMD converges faster (and smoother) to the final protonation
fraction.
While the convergence of protonation equilibrium is crucial for the proper computation
of a pKa, the convergence of structural properties is also important. To consider
this issue, we calculated the RMSD of α-Carbons for all pHs (for both CpH and pH-
-REMD simulations) with respect to the average structure from a CpH simulation at
pH 10. For both CpH and pH-REMD methods, the conformational convergence was
studied by calculation of the Kullback-Leibler divergence (see section 1.5.3.5) of RMSD
cumulative distributions vs. time. This is a measure of the rate of convergence to the
final conformational ensemble. We presented results for both CpH and pH-REMD at pH
10 for every 100 ps (Figure 4-7). We used the final RSMD distribution of CpH simulation
at pH 10 as a reference for the plot. As the inset of Figure 4-7 shows, both simulations
73
0
0.2
0.4
0.6
0.8
1
7 8 9 10 11 12 13
Dep
roto
natio
n F
ract
ion
pH
Lys5; CpH Lys5; pH-REMD
Tyr7; CpHTyr7; pH-REMD
Figure 4-5. The titration curves of Lys5 and Tyr7 in the heptapeptide derived fromOMTKY3.
converge to the same RMDS distribution. It is clear that pH-REMD converges more
rapidly and smoothly than CpH.
We also compared the rate of visiting distinct structures by calculating the RMSD
autocorrelation at all pH for both methods. According to Figure 4-8, the correlation
time of RMSDs in pH-REMD simulations are significantly shorter than that of CpH
simulations, which implies that pH-REMD visits distinct conformations more often than
CpH.
4.5 Concluding Remarks
In the present chapter, we have combined Hamiltonian REMD with Constant pH
MD to create what we call pH-REMD. The predicted pKa for a number of systems
using pH-REMD is in excellent agreement with experimental data. Compared to CpH
MD, pH-REMD converges faster in both conformational and protonation spaces. In
contrast to the Temperature Replica Exchange Molecular Dynamics methods (TREMD),
in pH-REMD the replica ladder (pH) is very limited, so the overlap of energy and
74
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10000 20000 30000 40000 50000 60000
Dep
roto
natio
n F
ract
ion
MC titration step
(A)
pH 8.0pH 9.0
pH 10.0pH 11.0pH 12.0pH 13.0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10000 20000 30000 40000 50000 60000
Dep
roto
natio
n F
ract
ion
MC titration step
(B)
pH 8.0pH 9.0
pH 10.0pH 11.0pH 12.0pH 13.0
Figure 4-6. Cumulative average protonation fraction for TYR7 versus MC titration stepsat pH=8.0,9.0,10.0,11.0,12.0 and 13.0, comparison between the CpH (A)and the pH-REMD (B) methods.
75
0
10
20
30
40
50
60
70
0 2 4 6 8 10
Kul
lbac
k-Le
ible
r di
verg
ence
t (Simulation Time (ns))
CpH; pH=10pH-REMD; pH=10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
3 3.5 4 4.5 5 5.5 6 6.5 7
RMSD (A)
Figure 4-7. The Kullback-Leibler divergence measure of RMSD distributions of CpH(Green) and pH-REMD (Red) with respect to the final RMSD distribution inCpH. The inset shows the RMSD distributions of CpH (Green) andpH-REMD (Red) respectively after 100 and 10 ns.
consequently the exchange ratio between neighboring replicas are always high.
This new method is expected to perform very well for biosystems with highly coupled
conformational and protonation states, like proteins with pH-dependent structure and
dynamics.
76
0
0.2
0.4
0.6
0.8
1
0 0.05 0.1 0.15 0.2
Aut
ocor
rela
tion
Lag (ns)
pH 2.0 pH 3.0 pH 4.0 pH 5.0
pH 6.0 pH 7.0 pH 8.0 pH 9.0
pH 10.0 pH 11.0 pH 12.0 pH 13.0
Figure 4-8. RMSD Autocorrelation for CpH (Green) and pH-REMD (Red) at all pHs.
77
CHAPTER 5OPTIMIZATION OF UMBRELLA SAMPLING REPLICA EXCHANGE MOLECULAR
DYNAMICS BY REPLICA POSITIONING
5.1 Literature Review
Among the advanced sampling methods (see section 1.4), the Umbrella Sampling
method (US)[25, 49, 170] is well known for improving the sampling of rare events along
the reaction coordinate. US attempts to solve the bad sampling problem by applying a
biasing potential to the system, to guarantee the effective sampling along the reaction
coordinates. This can be achieved either in one or several simulations (windows).
While the efficiency of US is clear, it needs careful tuning. For example, the positions of
windows and the strength of the bias potentials have to be such that they have enough
overlap.
REMD techniques [1–3, 22, 24] have become increasingly popular schemes. REMD
is not only a promising method for tackling the quasi-ergodicity[171] problem, but it
is also able to combine intuitively with other enhanced sampling techniques[2, 4, 21,
23, 24, 55, 168]. In REMD methods, N non-interacting copies (replicas) of the system
with different temperatures (Temperature REMD[2–4]) or different potential energy
parameters (Hamiltonian REMD[3, 22, 24, 56, 172]) run concurrently and each replica
attempts to exchange with its neighbors every few steps. In other words, the system
performs a random walk in the replicas’ ladder space.
Three types of efforts have been made to optimize REMD with most of those
being only applicable to TREMD. First, some techniques try to keep the number
of replicas limited. Second, there are methods that try to increase the efficiency of
REMD by increasing the probability of accepting an exchange. Third, are methods
that increase the efficiency by optimal selection of the positions of replicas on a
Temperature/Hamiltonian ladder. In TREMD, the number of needed replicas in a system
increases with the number degrees of freedom[56, 58, 173, 174]. This scaling confines
TREMD to small systems. A clear way to avoid this problem is to decrease the number
78
of degrees of freedom[175, 176]. Approaches comprise the use of hybrid explicit/implicit
solvent models for the MD and exchange parts respectively[173], the use of separate
baths for solvent and solute[174], the use of a coarse grained model for a subset of the
system[58] etc. Also, some studies have tried to decrease the number of replicas by
coupling the system to a pre-sampled reservoir[177, 178].
A few efforts have been made to increase the probability of exchange. Rick[179]
used dynamically scaled replicas between conventional replicas at broad temperature
ranges. Li et al. [180] devised a new approach called Temperature Intervals with Global
Energy Reassignment (TIGER). In their method, short runs of heating-sampling-
-quenching sequences are done and at the end of each sequence the potential energies
of all replicas are compared using Metropolis MC sampling and finally reassigned to
the temperature levels. Kamberaj et al.[181] added appropriate bias potentials to the
system to flatten the Potential of Mean Force and to achieve a random walk along the
temperature ladder. Also, Ballard et al. [168] used non-equilibrium works simulations to
generate the attempted configuration swaps.
For the optimal choice of the ladder of replicas, it has been argued that the highest
efficiency is achieved when the Exchange Acceptance Ratio (EAR) between neighbor
replicas is about 20%.[182–185]. In 2002, Kofke[182, 183] showed that when the
specific heat of the system is independent of temperature, assigning temperatures in
a geometric progression leads to equal EAR for TREMD. But, if the specific heat is a
function of temperature, then the choice of temperatures is not system-independent.
Although setting the equi-EAR between the neighbors is the most accepted
criterion for optimizing REMD (especially in TREMD); it has been shown [186–
190] that for systems with phase transitions along replica ladders, this is not the
optimal arrangement. A few techniques have been proposed for selecting the set of
temperatures when the specific heat changes with temperature. For example, Rathore
et al. [187] used a small set of pre-simulated replicas at different temperatures to
79
determine the mean and variance of energy at those temperatures and then applied
an iterative scheme to find the right positions of replicas (temperatures). Katzgraber
et al. [188] also devised an iterative feedback algorithm for optimizing TREMD, but
their method is computationally demanding. In 2007, Hirtz and Oostenbrink[189]
proposed a technique for optimizing TREMD and HREMD in systems with known
stable conformations. In their method, one first generates many short MD trajectories
using different Hamiltonians/temperatures and starting from different regions of
conformation space. Then one clusters the frames based on their conformational
states and Hamiltonians/temperatures at which those were simulated and finally applies
the transition probability between the pairs in different groups. Shenfeld et al. [191]
minimized the thermodynamic length [192] in order to find the optimal positions of
replicas.
The majority of proposed optimization methods for REMD are applicable only to
TREMD, with those that are applicable to HREMD methods being computationally
demanding. In this chapter we propose a method for optimizing Umbrella Sampling
Replica Exchange. We use three unique properties of USRE for optimizing the position
of replicas.
First, in the case of USRE, there is no phase transition along the reaction
coordinate, because the bias potentials constrain the sampled coordinate locally. As
described above, the optimal replica sets in the absence of phase transitions are those
that have equi-EAR between all neighbor replicas. Second, the exchange probability in
USRE is only a function of the difference between order parameters at the moment of
exchange. This Exchange Acceptance Ratio (EAR) is very easy to compute. Third, in
USRE we can safely approximate the order parameter distribution in each window as a
Gaussian distribution[27, 28]. In this chapter, we do not discuss or propose any optimal
value of EAR between the neighbors, but we provide a way to optimizes the efficiency of
USRE given either the number of replicas or the desired EAR between all neighbors.
80
To test our method we applied it to optimize USRE simulations to sampling butane
dihedral rotation in implicit solvent and to compute the free energy of formation of a
NH+4 + OH− salt bridge in explicit solvent. The proposed positions of replicas (windows)
by our method not only resulted in better mixing and convergence with respect to that of
equal distance on the reaction coordinate but also the best possible arrangement with
given range and number of replicas. We show that any deviation from this arrangement
produces a lower rate of mixing. In the rest of the chapter we use replica and windows
interchangeably.
5.2 Theory
5.2.1 Umbrella Sampling Replica Exchange
USRE[4] is a combination of Umbrella Sampling and Hamiltonian Replica Exchange
MD (see chapter 1), i.e., in USRE N non-interacting replicas run simultaneously with
N different bias potentials. The bias potentials are harmonic functions with distinct
strengths and coordinate references.
The bias of the ith window has a quadratic form of
Bi(ξ) = ki(ξ(q)− ξ0i )2, (5–1)
where ki, ξ(q), ξ0i , and q are the bias strength, the order parameter, the bias center
on the reaction coordinate, and configuration respectively. We will present results
only for the case for which all our replicas have the same force constant but different
coordinate centers. However the method is easily expandable to the case of dissimilar
bias strengths.
The Hamiltonian for the ith window can be written as:
Hi = K+UUB + Bi, (5–2)
where UUB and K are respectively the unbiased potential energy and the kinetic energy.
For two exchanging replicas, i and j, with ki = kj = k , ∆ in Equation 1–22 can be
81
simplified to:
∆ = β[k(ξi − ξ0j )
2 + k(ξj − ξ0i )2]−
[k(ξi − ξ0i )
2 + k(ξj − ξ0j )2]
= 2kβ(ξ0i − ξ0j )(ξi − ξj) = 2k∆ξ0ij∆ξij.
(5–3)
For a given choice of bias centers for each window, ∆ξ0ij is a constant and then ∆
becomes a linear function of ∆ξij and is not dependent on the potential energies of
exchanging replicas.
5.2.2 Calculating Exchange Acceptance Ratio in Umbrella Sampling ReplicaExchange
By inserting Equation 5–3 into Equation 1–20, we can write the probability of
accepting an exchange as:
Pacc = min(1, exp(−2βk ∆ξ0ij ∆ξij)
). (5–4)
Using the Probability Density Function (PDF) of ∆ξij (i.e., p(∆ξij), the EARij (or the
average probability of acceptance between replicas i and j) can be calculated as:
EARij = ⟨Pacc⟩p(∆ξij)=
∫Pacc(∆ξij)p(∆ξij)d∆ξij. (5–5)
We can estimate the p(∆ξij)as a normal Distribution (see appendix A for details)
p(∆ξij) =1√
2π(σ2ξi+ σ2
ξj
) exp
−(∆ξij − (⟨ξi⟩ − ⟨ξj⟩))2
2(σ2ξi+ σ2
ξj
) . (5–6)
here, ⟨ξi⟩ and σ2ξi
are the mean and the variance of ξi distribution. Note that ⟨ξi⟩ = ξ0i .
By substituting the Equation 5–4 and Equation 5–6 in the Equation 5–5, the average
probability of acceptance becomes
EARij =
∫ ∞
−∞min
(1, exp(−2βk ∆ξ0ij ∆ξij)
) 1√2π
(σ2ξi+ σ2
ξj
) exp
−(∆ξij − (⟨ξi⟩ − ⟨ξj⟩))2
2(σ2ξi+ σ2
ξj
) d∆ξij.
(5–7)
82
Without loss of generality, we can simplify the min by assuming ξ0i < ξ0j , and finally
EARij =1
2erfc
− (⟨ξi⟩ − ⟨ξj⟩)√2(σ2ξi+ σ2
ξj
)
+1
2erfc
(⟨ξi⟩ − ⟨ξj⟩)− 2βk(∆ξ0ij)(σ2ξi+ σ2
ξj
)√
2(σ2ξi+ σ2
ξj
)
exp[−2βk(∆ξ0ij)(⟨ξi⟩ − ⟨ξj⟩) + 2β2k2
(σ2ξi+ σ2
ξj
)(∆ξ0ij)
2].
(5–8)
In Equation 5–8, all other parameters except the ⟨ξi|j⟩ and σ2ξi|j
are known. In appendix
B we present a method for estimating the mean and variance of ξ at any point of the
reaction coordinate using very few short pre-simulated windows.
5.2.3 Umbrella Sampling Replica Exchange Optimization
As we discussed above, there is no phase transition along the replicas ladder in the
case of USRE. If there is no phase transition, the best solution for positions of replicas
is for the EAR between all neighbor pairs is equal. We present here two different
approaches for making a set of equi-EARs among neighbor pairs. We discuss fitness
functions for each case, that, when maximized, lead to equal EAR among neighbor
replicas.
In the first method, the number of replicas and the center of the first and last
replicas are fixed (the range for the sampling coordinate). In order to find the set of
equi-EAR replicas, we define a Full Batch Scoring Function (FBSF) as:
FBSF(ξ02 ... ξ
0N−1
)= −
N∑i=2
(EARi−1i(ξ
0i−1 − ξ0i )− ⟨EAR⟩
)2 (5–9)
and maximize it with respect to the center of biases in all replicas except the last
and first ones (i.e., ξ02 ... ξN−12 ). Here N is the number of replicas and ⟨EAR⟩ =
N∑i=2
EARi−1i(ξ0i−1−ξ0i )
N−1.
83
In FBSF we set the position of the first and the last replicas. Consequently the total
number of degrees of freedom is equal to N − 2. Since EARi−1i is a strictly decreasing
function of ξ0i−1 − ξ0i , then it can be concluded that the scoring function (which is an N− 2
dimensional parabola) has a unique maximum. Maximizing the FBSF leads to equal
EAR between adjacent replicas.
In the second method, we choose a desired value for the EAR between replicas,
and we fix the center of bias for the first replica (lowest value on reaction coordinate)
and the range of the reaction coordinate. In this scheme, starting from the lowest value
on the reaction coordinate, we add a new replica at a time and maximize the Single
Batch Scoring Function (SBSF) respect to that center of bias in that replica. We continue
this scheme until the last replica passes the maximum range of the reaction coordinate.
In order to adjust the position ith replica, we try to maximize the following SBSF at every
replica accumulation:
SBSF(ξ0i)= −
(EARi−1i(ξ
0i−1 − ξ0i )− EARc
)2 (5–10)
where EARc is the desired value of EAR, which is bounded between 0 and 1. At the end
of this process the EAR between all neighbor pairs equals EARc. We used the Barzilai
and Borwein[194, 195] optimization method for maximizing both SBSF and FBSF (see
the Appendix C for more details).
5.2.4 Umbrella Sampling Replica Exchange Optimization Workflows
Based on the choice of optimization, two types of workflows exist. In both methods
we pre-simulate M windows. In the FBSF method, the number of replicas is set, but the
final average EAR is not known a priori (Figure 5-1). In the SBSF method the EARc is a
chosen parameter and the number of replicas in not known in advance (Figure 5-2).
We have written scripts (for both FBSF and SBSF) that implement the ideas
presented here. Both scripts are available at http://www.clas.ufl.edu/users/
roitberg/software.html.
84
• pre-simulated data from M windows• convergence criterion (ǫ)• N (Number of windows)• range of reaction coordinate
calculate the meansand variancesof M windows
initilize N equi-distance center ofbiases along the
reaction coordinate
estimate the meanand varianceof N windows
estimate the EARsof N windows
|FBSF |<ǫ
stop
move all replicas alongthe FBSF gradient
yes
no
Figure 5-1. FBSF workflow
85
• pre-simulated data from M windows• convergence criterion (ǫ)• Exchange Acceptance Ratio (EARc)• range of reaction coordiante
estimate the meansand variancesof M windows
adding the first replica
adding thenext replica
estimating the EARwith previous replica
|SBSF |<ǫ
is thepositionof the lastreplica isin thegivenrange?
stop
move the lastreplica along theSBSF gradient
no
yes
no
yes
Figure 5-2. SBSF workflow
5.2.5 Simulation Details
In order to validate our method, we performed USRE simulations on two systems.
First, we studied the PMF along the butane torsional angle in the range from -65 to
67 degrees using 12 replicas. We performed four sets of simulations for different
arrangements of replicas on the reaction coordinate. We applied the OBC Generalized
Born[101] implicit solvent model (igb=5) with a cutoff of 12 A for nonbonded interactions.
A spring constant of k = 30kcalmol−1radian−2 has been applied to all replicas.
The length of each simulation was about 50 ns in order to have enough sampling
86
for estimating the Average Roundtrip Time between the highest and the lowest
replicas in each set. Second, we performed two sets of simulations using 24 replicas
on a NH+4 + OH− salt bridge immersed in a box of TIP3P solvent model[196] of
21.4×21.4×21.4 A3. We calculated the long range electrostatic interactions with the
particle mesh Ewald method, and used an 8A cutoff for the short range nonbonded
interactions. A spring constant of k = 30kcalmol−1A−2 has been applied to all replicas.
The lengths of all simulations were 40 ps, however we used a 5 ns USRE simulation as a
reference in calculation of the Kullback-Leibler divergence[39].
In all calculations in this chapter, we used the AMBER 12 molecular simulation
suite[197] and the general AMBER force field (GAFF)[198]. The SHAKE[77] algorithm
was employed to constrain the distance between hydrogens and heavy atoms, which
permitted the use of 2 fs MD steps. Replica were attempted to exchange every 50 MD
steps[199] and Langevin dynamics with friction coefficient of 2.0 ps−1 was employed
to sustain the systems at 300K. In order to produce independent simulations[107],
we used distinct random seeds (ig=-1 in AMBER) for the Langevin thermostats in all
simulations. We saved the order parameter every 10 MD steps (i.e., 20 fs) and we used
Grossfields[26] implementation of Weighted Histogram Analysis method (WHAM) for
calculating the Potential of Mean Forces versus reaction coordinate in all simulations.
5.3 Results and Discussion
To study the effect of replica optimization in USRE, we computed and compared the
Average Roundtrip Time for different positions of the replicas, including our optimized
one. Furthermore we measured the convergence speed for optimized and non-
-optimized USRE arrangement of NH+4 + OH− salt bridge in explicit solvent. In both
simulations, we choose the total number of replicas to be equal to that of pre-simulated
replicas (i.e., M = N).
87
5.3.1 Potential of Mean Force Along the Butane Dihedral Angle
We performed four sets of USRE calculations on butane dihedral in implicit solvent.
In the first one, the centers of biases were equally distributed (every 12) on the reaction
coordinate. In the second set of calculations we moved two neighboring replicas in the
middle of the reaction coordinate and we kept all other replicas in place. For the third
set we optimized the replica positions by maximizing the FBSF. For the optimization,
we used the first 50 ps of the first set as pre-simulated data. Finally, in the fourth set of
simulations we perturbed the position of one replica in the third (optimized) set. The
positions of replicas and corresponding EAR in each set has been shown in Table 5-1.
Table 5-1. Position and EAR for four different settings of USRE calculations on thebutane dihedral simulation in implicit water. Position of each replicacorresponds to the center of bias for that replica. Also, the EAR of eachposition corresponds to the EAR between that replica position and its higherneighbor on the reaction coordinate.
Set 1 Set 2 Set 3 Set 4Position EAR Position EAR Position EAR Position EAR
-65.00 0.18 -65.00 0.19 -65.00 0.14 -65.00 0.14-53.00 0.18 -53.00 0.18 -51.72 0.14 -51.72 0.14-41.00 0.16 -41.00 0.16 -38.58 0.14 -38.58 0.14-29.00 0.13 -29.00 0.13 -26.01 0.14 -26.01 0.14-17.00 0.10 -17.00 0.13 -14.44 0.15 -14.44 0.15
-5.00 0.07 -6.00 0.03 -4.16 0.14 -4.16 0.107.00 0.10 9.00 0.18 5.73 0.15 7.00 0.21
19.00 0.14 19.00 0.14 16.10 0.14 16.10 0.1331.00 0.16 31.00 0.17 27.98 0.14 27.98 0.1443.00 0.18 43.00 0.18 40.60 0.14 40.60 0.1455.00 0.19 55.00 0.19 53.71 0.14 53.71 0.1467.00 0.00 67.00 0.00 67.00 0.00 67.00 0.00
For each set, the Average Roundtrip Time (ART) between extremum replicas was
computed. To gather sufficient statistics, each simulation was repeated 10 times. Since
there were 12 replicas in each simulation, the total number of measured ARTs in each
set was 120. We present below the probability density of the ARTs (Figure 5-3). Figure
5-3 can be analyzed based on the Central Limit Theorem. The CLT describes the
characteristics of the population of the means of an infinite number of random samples
88
800 900 1000 1100 1200 1300 1400
0.000
0.005
0.010
0.015
ART(Exchange)
De
nsi
ty
set 1
set 2
set 3
set 4
−60 −40 −20 0 20 40 60
01
23
4
ξ
PM
F(k
cal/
mo
l)
Figure 5-3. ART probability density. The inset shows the PMF vs. reaction coordinate(dihedral angle)
of size N, drawn from an infinite parent population. According to the CLT, the distribution
of the mean converges to a Gaussian distribution, and the mean of the means (i.e.,
center of the Gaussian) is equal to the mean of the parent distribution.
Based on the CLT the centers of distributions are equal to the mean of the parent
pools. According to Figure 5-3, the Optimized set (set 3) has the lowest ART among
the all sets, moreover as the replica positions deviate more from optimized arrangement
(i.e., sets 4, 1 and 2 respectively) the ART increases. This shows the equi-EAR set
corresponds to the minimum ART and the mixing is the best in this arrangement.
Since the number of exchange attempts is equal in all sets, we expect the sets with
higher ARTs to have the smaller sample sizes (i.e., the number of roundtrips in each
simulation), which according to the CLT leads to wider ART distributions. Moreover,
Figure 5-3 also shows the naive arrangement of equal distance between the centers of
biases (set 1) is not the optimized arrangement of replicas.
89
5.3.2 Potential of Mean Force for NH+4 +OH− Salt Bridge in Explicit Solvent
In order to show the effect of USRE optimization on the rate of the convergence for
free energy calculations, we applied it to a more computationally demanding system,
i.e., NH+4 + OH− salt bridge in explicit solvent. We have computed the salt bridge PMF
vs. N−O distance (inset of Figure 5-5).
We performed 2 sets of simulations on the salt bridge and for each set we repeated
the calculations 10 times. In set 1 we used the naive setting of equal distance between
the adjacent bias centers along the reaction coordinate and in the set 2 we optimized
the positions of replicas. For the optimization leading to set 2, we used dumped values
of the first 20 ps of one of the simulations in set 1 to maximize the FBSF.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45equi-distance
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
2.5 3 3.5 4 4.5 5 5.5 6 6.5 7
EA
R
Replica Position (ξ)
optimized
Figure 5-4. Window positions vs. EARs in the equi-distance (non-optimized) (red circles)and optimized (set 2) (green circles) simulations. The EAR of each positioncorresponds to EAR between that replica position and its higher neighbor onthe reaction coordinate.
90
0
0.002
0.004
0.006
0.008
0.01
0.012
0 5 10 15 20 25 30 35 40
< K
ullb
ack−
Leib
ler
dive
rgen
ce >
Time (ps)
set 1 (non−optimized)set 2 (optimized)
0
1
2
3
4
5
3 4 5 6 7
PM
F (
kcal
/mol
)
ξ (N−O distance, Angstrom)
reference PMF
Figure 5-5. The mean and RMSE of Kullback-Leibler divergence over 10 simulations foreach set. Set 1 (red) is the naive equi-distance arrangement and set 2(green) is our optimized arrangement. We used the PMF of 5 ns USREsimulation as a reference for both simulation sets (the inset). The insetshows PMF vs. reaction coordinate (N−O distance)
5.4 Concluding Remarks
In the present work, we used statistical mechanics techniques to estimate the
Exchange Acceptance Ratios between any two windows on the reaction coordinate for
USRE.
Using the proposed schemes we set the positions of replicas such that all replicas
have the same EAR with their immediate neighbors. Since in USRE there is no phase
transition along the reaction coordinate then this setup maximizes the number of
roundtrips between the lowest and highest replicas and consequently the efficiency of
USRE. We applied our optimization method to butane in implicit solvent and NH+4 +OH−
salt bridge in explicit solvent. From the results, it was evident that this optimization
substantially increased the mixing and convergence rate for both systems. We expect
91
the optimization of USRE to significantly increase the speed of convergence in larger
systems.
92
CHAPTER 6CONCLUSIONS
In spite of the excessive growth in the power of computer hardware in recent years,
still the role of efficient sampling in Computational biophysics and biochemistry is
crucial. My PhD research was mostly dedicated to developing and optimizing enhanced
sampling methods and predominantly Hamiltonian Replica Exchange Molecular
Dynamics (HREMD). In my first project, we demonstrated that the HREMD method
improves the convergence rate in alchemical free energy calculations. Moreover,
we showed that there is a direct mapping between the HREMD and Free Energy
Perturbation (FEP) methods, which can be used for both exchange acceptance and free
energy calculations.
In the second project, I used the Replica Exchange Free Energy Method to
estimate the pKa shift in Glutamate 66 in a hyperstable mutant of staphylococcal
nuclease. In addition, we aimed to resolve an inconsistency between pKa experimental
and continuum methods in estimation of pKa value of position 66 in that protein. We
proposed that the experimental methods, which are mostly sensitive to configurational
changes, measure the equilibrium constant between two configurational states
instead of two protonation states. The results are in almost perfect agreement with
the Morenoes lab experiments.
In the third project, we developed and validated a pH-Replica Exchange Molecular
Dynamics (pH-REMD) method. We applied this method to a series of model compounds,
a terminally charged ADFDA pentapeptide, and a heptapeptide derived from the
Ovomucoid third domain (OMTKY3). We showed that not only the predicted pKas in
pH-REMD are very similar to that of Constant pH MD (CpHMD), but also the sampling is
more effective than that of CpHMD methods.
Finally in the last project, we focused on optimizing the Umbrella Sampling Replica
Exchange (USRE) method. We invented, validated, and tested a method for estimating
93
the probability of exchange between neighboring replicas. Using information from
short umbrella runs, we optimized the position of replicas (windows) along the reaction
coordinate, and we showed that the equal exchange acceptance between replica pairs
is the optimum setting for USRE.
94
APPENDIX ACALCULATING THE PDF OF ∆ξ IN USRE
In US (and subsequently USRE), approximating the Probability Density Function
(PDF) of ξ by a normal distribution is valid assumption[169]. So e.g., for replica i we can
write:
pi(ξ) =1√2πσ2
ξi
exp
[−(ξ − ⟨ξi⟩)2
2 σ2ξi
]. (A–1)
Since the windows are independent, the joint PDF of ξi and ξj (i.e., p(ξi, ξj)) is equal to
multiplication of their PDF; then we can compute the PDF of ∆ξij = ξi − ξj by integrating
the joint PDF of ξi and ξj, i.e.,
p(∆ξij) =
∫ ∞
−∞dξi p(ξi, ξj) =
∫ ∞
−∞dξi p(ξi) p(ξj) =
∫ ∞
−∞dξi p(ξi) p(−∆ξij + ξi)
=
∫ ∞
−∞
1
2πσξiσξj
exp−1
2
[(ξi − ⟨ξi⟩)2
σ2ξi
+(−∆ξij + ξi − ⟨ξj⟩)2
σ2ξj
]d∆ξi
=1√
2π(σ2ξi+ σ2
ξj
) exp
−(∆ξij − (⟨ξi⟩ − ⟨ξj⟩))2
2(σ2ξi+ σ2
ξj
) .
(A–2)
which itself is a normal distribution.
95
APPENDIX BESTIMATING MEANS AND VARIANCES OF THE WINDOWS USING NEAREST
NEIGHBOR WEIGHTED AVERAGING AND REWEIGTHING
In order to find the optimal positions of replicas on the reaction coordinate, we need
to estimate the mean and variance at the replicas’ new positions.
The most primitive way to estimate the mean and variance of ξ (i.e. ⟨ξ⟩ and σ2)
for each window is to run a short US simulation at that window. Since optimizing the
positions of replicas requires moving the positions (i.e., center of biases) of replicas at
each step of optimization, it is not very efficient to use this method for estimating the
mean and variance of ξ. Instead, we develop a method below, which approximates ⟨ξ⟩
and σ2 for any arbitrary replica on the reaction coordinate using the ⟨ξ⟩ and σ2 of a few,
very short time pre-simulated windows.
Suppose we know the center of bias, variance and mean for N pre-simulated
windows, i.e., ξ01 ... ξ0N
⟨ξ1⟩ ... ⟨ξN⟩
⟨ξ21⟩ ... ⟨ξ2N⟩
. (B–1)
If replica i is a pre-simulated replica, then theoretically we can estimate the first two
non-central moments of the ξ distribution (i.e., ⟨ξ⟩ and σ2) on any new window m on the
reaction coordinate using the reweighting technique (section B.1 )as:
⟨ξ⟩m;i =
∫∞−∞ ξ 1√
2πσ2ξi
exp
[− (ξ−⟨ξi⟩)2
2σ2ξi
]exp [βk ((ξ − ξ0i )
2 − (ξ − ξ0m)2)] dξ
∫∞−∞
1√2πσ2
ξi
exp
[− (ξ−⟨ξi⟩)2
2σ2ξi
]exp [βk ((ξ − ξ0i )
2 − (ξ − ξ0m)2)] dξ
(B–2)
and
⟨ξ2⟩m;i =
∫∞−∞ ξ2 1√
2πσ2ξi
exp
[− (ξ−⟨ξi⟩)2
2σ2ξi
]exp [βk ((ξ − ξ0i )
2 − (ξ − ξ0m)2)] dξ
∫∞−∞
1√2πσ2
ξi
exp
[− (ξ−⟨ξi⟩)2
2σ2ξi
]exp [βk ((ξ − ξ0i )
2 − (ξ − ξ0m)2)] dξ
(B–3)
96
where ⟨f(ξ)⟩m;i is the average of f(ξ) over distribution of m using the distribution of i. We
can solve the above integrations by writing the power of exponential in a single square
form:
−(ξ−⟨ξi⟩)2
2σ2ξi
+ βk((ξ − ξ0i )
2 − (ξ − ξ0m)2)
=[−2⟨ξi⟩βk(ξ0i − ξ0m) + 2σ2
i (βk(ξ0i − ξ0m))
2]−
(ξ − (⟨ξi⟩ − 2σ2i βk(ξ
0i − ξ0m))
2σ2i
+ βk((ξ0i )
2 − (ξ0m)2) (B–4)
and using ∞∫
−∞
1√πC
ξ exp[− (ξ−B)2
C
]= B
∞∫−∞
1√πC
ξ2 exp[− (ξ−B)2
C
]= 1
2C + B2
, (B–5)
Equations (B–2) and B–3 simplifies to: ⟨ξ⟩m;i = ⟨ξi⟩ − 2σ2i βk(ξ
0i − ξ0m)
⟨ξ2⟩m;i = (⟨ξi⟩ − 2σ2i βk(ξ
0i − ξ0m))
2+ σ2
i
, (B–6)
If replicas i and i + 1 are the nearest pre-simulated neighbors of our imaginary replica m,
i.e., ξ0i < ξ0m < ξ0i+1, then using the nearest neighbor weighted averaging method (section
B.2) we can estimate the mean and variance of ξ on windows m as following:⟨ξ⟩m = wi (⟨ξi⟩ − 2σ2
i βk(ξ0i − ξ0m)) + wi+1
(⟨xii+1 − 2σ2
i+1βk(ξ0i+1 − ξ0m)
)⟨ξ2⟩m = wi
[(⟨ξi⟩ − 2σ2
i βk(ξ0i − ξ0m))
2+ σ2
i
]+ wi+1
[(⟨ξi+1⟩ − 2σ2
i+1βk(ξ0i+1 − ξ0m)
)2+ σ2
i+1
]σ2m = ⟨ξ2⟩m − ⟨ξ⟩2m
,
(B–7)
97
where we choose the power parameter equal to 1.2 and number of involving nearest
neighbors equals to 2, so Equation B–14 in the section B.2 implies:
wi =
(|ξ0i −ξ0m|)
0.6
(|ξ0i −ξ0m|)0.6
+(|ξ0i+1−ξ0m|)0.6 ξ0m = ξ0i and ξ0m = ξ0i+1
1 ξ0m = ξ0i
0 ξ0m = ξ0i+1
. (B–8)
Using the above setup, we are able to estimate the mean and variance of any
arbitrary window on the reaction coordinate. Subsequently it is possible to approximate
the EAR between any two replicas using Equation 5–8.
B.1 Reweighting Fitted Gaussian Distribution on Windows
Here we describe the reweighting method that we used for estimation of the mean
and the variance of arbitrary windows on reaction coordinate. Consider a continuous
function f(ξ). We can calculate the average of f(ξ) over a distribution of p(ξ) as following:
⟨f(ξ)⟩p =
∫f(ξ) p(ξ) dξ∫p(ξ) dξ
. (B–9)
If we want to calculate the average of f(ξ) over another distribution q(ξ), where the
domain of q(ξ) is a subset of p(ξ), then we may write:
⟨f(ξ)⟩q;p =
∫f(ξ) q(ξ) dξ∫q(ξ) dξ
=
∫f(ξ) p(ξ)q(ξ)
p(ξ)dξ∫
p(ξ)q(ξ)p(ξ)
dξ=⟨q(ξ)p(ξ)
f(ξ)⟩p⟨q(ξ)p(ξ)⟩p
, (B–10)
where ⟨f(ξ)⟩q;p is the average of f(ξ) over distribution of q using the distribution of p.
In the case of Umbrella Sampling and by assuming Gaussian distributions for the
collective variables for every window (i.e., using Equation A–1), we can estimate the
average of f(ξ) over window i:
⟨f(ξ)⟩pi =
∫f(ξ) 1√
2πσ2ξi
exp
[− (ξ−⟨ξi⟩)2
2 σ2ξi
]dξ
∫1√
2πσ2ξi
exp
[− (ξ−⟨ξi⟩)2
2 σ2ξi
]dξ
. (B–11)
98
Using the Equation 5–1 and Equation 5–2, we can compute the ratio of probability
distribution of replicas i and m as
pm(ξ)
pi(ξ)=
exp [−β(UUB + Bm)]
exp [−β(UUB + Bi)]= exp
[βk
((ξ − ξ0i )
2 − (ξ − ξ0m)2)]
(B–12)
Finally using Equation B–9 and Equation B–12, we can exploit
⟨f(ξ)⟩pm;pi = ⟨f(ξ)⟩m;i =
∫f(ξ) 1√
2πσ2ξi
exp
[− (ξ−⟨ξi⟩)2
2 σ2ξi
]exp [βk ((ξ − ξ0i )
2 − (ξ − ξ0m)2)] dξ
∫1√
2πσ2ξi
exp
[− (ξ−⟨ξi⟩)2
2 σ2ξi
]exp [βk ((ξ − ξ0i )
2 − (ξ − ξ0m)2)] dξ
.
(B–13)
Which is the average of f(ξ) over the window m using the window i. Using Equation
B–13 we calculate the values of ⟨ξ⟩ and ⟨ξ2⟩ for any arbitrary windows, using the pre-
-simulated windows (i.e., Equation B–6).
B.2 Nearest Neighbor Weighted Averaging (NNWA)
We used NNWA to make a better estimation of mean and variance of ξ for any
arbitrary window by pre-simulated windows. NNWA is a member of Locally Weighted
Learning methods[200]. Consider a function f(ξ) with a few known points (i.e., ξi);
furthermore suppose the value of f at any arbitrary ξ position can be estimated
separately by each of those known points; then using k known nearest neighbor of
ξ, we can approximate the value of f, as:
f(ξ) =k∑
i=1
wi(ξ) fξi(ξ)k∑
i=1
wi(ξ). (B–14)
where fξi(ξ) is the estimation of f(ξ) using ξi and wi(ξ) = 1d(ξ,ξi)
. d is so-called metric
operator, which in its simplest form is d(ξ, ξi) = [(ξ − ξi)2]. p is a positive real number
known as a power parameter. Depending on the problem, k can be changed from the
first nearest neighbors to the total number of known points. In this paper we set p = 1.2
and k = 2, which helped the optimizer to smoothly converge.
99
APPENDIX CBARZILAI AND BORWEIN OPTIMIZATION (BB METHOD)
The Steepest Descent or Gradient Descent method is among the most important
first order optimization algorithms. It can be formulized as follows:
xk+1 = xk + λk gk, (C–1)
where gk = −∇f(xk) and λk is the step size at step k. There are many methods
for estimating the optimal step size[195, 201]. BB method[194] is one of the most
efficient routines among those, in which the step size is determined by minimizing either
∥∆x−λ∆g∥2 or its correspondent ∥λ−1∆x−∆g∥2 with respect to λ, where ∆x = xk−xk+1
and ∆g = gk − gk+1. Based on those, the optimal step size can be derived as:
λk =∆x⊺∆g
∆g⊺∆g. (C–2)
100
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BIOGRAPHICAL SKETCH
Danial Sabri Dashti was born in, Tehran, Iran. He went to the University of Kashan
majoring in physics and graduated with a bachelor’s degree in 2003. Then, he went to
Sharif University of Technology where, he received his master’s degree in physics in
2006. After finishing his master, he moved to Gainesville, Florida in 2007. He entered
the Department of Physics of the University of Florida to pursue a Ph.D degree in
computational biophysics. He received his Ph.D. from the University of Florida in the
summer of 2013.
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