Modeling of Biochemical Reactionsusers.ece.cmu.edu/~brunos/Lecture2.pdf · Modeling of Biochemical...

Dr. Carlo Cosentino Carnegie Mellon University, Pittsburgh, 20081

Modeling of Biochemical Reactions

Dr. Carlo CosentinoSchool of Computer and Biomedical Engineering

Department of Experimental and Clinical MedicineUniversità degli Studi Magna Graecia

Catanzaro, [email protected]

http://bioingegneria.unicz.it/~cosentino


Outline

Modeling of biochemical reactions

Deterministic models

Michaelis–Menten model

The Quasi–Steady–State Approximation

Allosteric reaction

Regulation of enzymatic reactions

Stochastic models

Stochastic derivation

The Gillespie algorithm


Biochemical Energy

The equilibrium of a reaction is linked to the variation of biochemical standard free energy, ∆G’0

The velocity, instead, depends on the activation energy, ‡PSG →∆


Reaction Rate

The reaction rate is determined by

The concentration of the reactants

The kinetic constant, usually denoted by k

The reaction rate for is

From transition-state theory it is possible to derive the relation

PS k⎯→⎯

[ ]SkV =

RTGehTk

‡∆−=k k: Boltzmann constant

h: Planck constant


Reaction Equilibrium

Let us consider the following basic reversible reaction

It can be described by the system

The equilibrium constant is a function of ∆G’0

PSPS

k

k

⎯⎯←⎯→⎯

−1

1

[ ] [ ] [ ][ ] [ ] [ ]PkSkdtPd

PkSkdtSd

11

11

−

−

−=

+−= [ ][ ] 1

1

−

==′kk

SPKeq

eqKRTG ′−=°′∆ ln R=8.315 J/mol·K (gas constant)

T=temperature [K]


Enzymatic Reaction Kinetics

Enzymes are a family of proteins specialized in the catalysis of reactions

Catalysts do not react and do not affect the reaction equilibrium, however they increase the reaction rate by decreasing the activation energy

The basic mechanism is the formation of an enzyme-substrate complex, which creates a more favorable condition for the formation of the product

After the substrate has been transformed into product, the complex dissociates and the enzyme is free to catalyze the next reaction


The Role of Enzymes

Enzymes play a central role in all the biological processes (metabolism, regulation, signaling)

Many diseases are caused by deficiency of some enzyme

Many drugs act by interacting with enzymes


Outline





Allosteric reaction


Stochastic models




Michaelis-Menten Model

The basic model of enzymatic reaction was proposed by Michaelis and Menten in 1913

The law of mass action (LMA) states that the reaction rate is proportional to the product of the concentrations of the reactants

L. Michaelis (1875-1949)

M. Menten (1879-1960)

CBA k⎯→⎯+[ ] [ ][ ]BAkdtCd

=

mA+ nBk−→ C

d[C]

dt= k[A]m[B]n


Michaelis-Menten Model (cont’d)

Applying the LMA to the model reaction scheme, we obtain

with initial conditions s(0)=s0, e(0)=e0, c(0)=0, p(0)=0

Note that the equation of p(t) is decoupled and yields

( )( ) ckpckkeskc

ckkeskeckesks

2211

21111

,,

=+−=++−=+−=

−

−−

&&

&&

[ ] [ ] [ ]PpEScEeSs ==== :,:],[:,:

( ) ( )∫=t

dcktp0

2 ττ


Simplified M-M Model

The total amount of enzyme (free+bound) remains constant over time, indeed, summing the equations of e(t) and c(t)

from which we can derive e(t) and substitute into the other eqs, obtaining

( ) ( ) 00 etctece =+⇒=+ &&

( ) ( )( ) ( ) 00,

0,

21101

01101

=++−==++−=

−

−

cckksksekcssckskseks

&

&


Simplified M–M Model (cont’d)

Typically, the initial formation of the complex ES is much faster than the product formation, hence it can be assumed to be instantaneous

This assumption lets us pose dc/dt ≅ 0, that is

where the positive constant

is the Michaelis–Menten constant

( ) ( )( ) mm Ks

seks

Ktstse

tc+

−=⇒+

= 020 &

1

21

kkkKm

+= −


Simplified M–M Model (cont’d)

Furthermore, the amount of enzyme is typically much less than that of substrate (at least in metabolic reactions), so one can assume that s(0)=s0also in the simplified model


Adimensionalization

A change of variables is often used in biochemical models, in order to obtain adimensional normalized parameters

This helps in analyzing the behavior of the system for different values of the parameters

Furthermore, the adimensionalization typically reduces the number of parameters

For the M–M reaction we can use the following change of variables

( ) ( ) ( ) ( )

0

0

001

21

01

2

0001

,,

,,,

se

sK

skkkK

skk

etcv

stsutek

m ==+

==

===

− ελ

τττ


Adimensional M–M model

By applying the change of variable above, the simplified model reduces to

If the amount of enzyme is much less than that of substrate, then ε¿ 1

( ) ( )( ) ( ) 00,

10,=+−==−++−=

vvKuuvuvKuuu

&

&

ελ


Analysis of the Time Constants

Assuming that s0 does not undergo a significant variation during the initial transient, it is possible to evaluate the length of such interval by the eq.

The time constant of this first-order system is

An estimate for the substrate transformation time constant can be derived by using the maximum derivative

( )cKsksekc m+−= 01001&

( )mc Ksk

t+

=01

1

02

0

max

0

ekKs

dtdss

t ms

+≈≈


Analysis of the Time Constants

Finally, we can give analytical conditions for the validity of the simplified model in terms of tc and ts

The amount of s consumed during the initial transient has to be negligible, that means |∆ s/s0|¿ 1

Actually, even in the case when e0/s0=O(1), the latter assumption is satisfied for large values of Km (that is when the reaction is slow)

( )12

01

02 <<+ mKskek

10

0 <<+

=mKs

eε


Experimental Parameters

In the experimental practice, not all the kinetic parameters of the reaction are measured, but rather

the M–M constant, Km

the maximum reaction rate,

[ ] 02max0 ekRQ ==

mm KsQs

Kssek

R+

=+

=0

0

0

0020


Outline





Allosteric reaction


Stochastic models




Quasi–Steady–State Approximation

The assumptions exploited in the derivation of the simplified M–M model take the name of quasi steady-state approximation (QSSA)

Large differences in the time-scales of the reactions may create huge difficulties both in terms of simulation and of understanding the basic principles of operation

To overcome this limitations, theoreticians (especially in the biochemistry community) often use QSSA to eliminate the fastest (and the slowest) equations in the system of ODE


Validity of the QSSA

The validity of the QSSA depends on

How large is the difference in the time–scales of the reactions

How large is the difference between the amount of enzyme and that of substrate

In the case of protein interaction networks (PINs) the QSSA assumptions are not satisfied, indeed

Enzymes have multiple substrates

Substrates are acted upon by multiple enzymes

Enzymes and substrates often swap roles (e.g. two kinases can phosphorylate each other)


Use and Abuse of the QSSA

An enlightening work concerning the validity of the QSSA

The authors performed a compared analysis of the Van Slyke–Cullen mechanism, a special case of the M–M reaction, with and without applying the QSSA

E.H. Flach, S. Schnell, Use and abuse of the quasi-steady-state approximation, IEE Proc.–Syst. Biol. 153(4), 187–191, 2006


Van Slyke–Cullen Model

By exploiting the conservation of the total amount of enzyme, we obtain

Which can be rescaled, in order to get rid of the parameters k1 and k2, by applying the change of variable

In the next slides we will not make use of the bars to refer to the new variables


Consideration on the Fluxes

The mass is conserved within the system over time, therefore

If v is constant the system can be reduced to a second-order system

When vi=0, the system is said to be in closed form, since there is neither input nor output

If v1(p) is constant, the equation of p(t) is decoupled (as in the M–M model)

s+ c+ p = v = v1 − v2

s = v1(p)− k1s(e0 − c)c = k1s(e0 − c)− k2c


Stability Analysis of the Closed System

Having a second-order system, it is possible to visualize the trajectories on a phase-plane

The first step consists of finding the null surfaces, by setting to zero the derivatives, which yields

The intersection of the null surfaces gives the steady state

which depends on k1 and k2, as can be seen by substituting for the original state variables


Stability Analysis of the Simplified Model

Once the equilibrium point has been computed, analysis of the linearized system provides information about the local stability in the neighborhood of that point

In the case of constant v the eigenvalues of the linearized system are real and negative; this corresponds to a so–called stable–node in the phase plane

c = 0 surface

The trajectories are attracted by a slow invariant manifold, which is confined to the region bounded between the null surfaces (the null surface of s is not shown)


Stability Analysis of the Full Model

We can repeat the stability analysis for the full third–order model, in order to check if the two models have always the same behavior

In this case, linearization around the equilibrium point and computation of the eigenvalues can lead to different cases:

The system is still locally asymptotically stable

For certain values of v1 and v2, complex eigenvalues arise, leading to a so–called stable focus in the phase plane

c = 0 surface


QSSA Model of the Open System

We will now assume v1(p) ≠ v2(p), that is the open system

Assume that, after the initial transient, the amount of complex changes very slowly, such that dc/dt≈ 0

It is therefore possible to express c as a function of s

Then, substituting in the other equations, we get

s ≈ v1(p)−e0s

1+ s

p ≈ e0s

1+ s− v2(p)


Comparison of QSSA and Full Model

Also in this case the systems behavior can be very different

The two phase plane below have been obtained using the same parameters and fluxes, with the full model (a) and QSSA model (b)

The full model exhibit a limit cycle, whereas the trajectories of the reduced one follow a spiral stable mode


QSSA is not Always Reliable

The QSSA is probably the most frequently used method for reducing the complexity of biochemical pathways models

Nonetheless it has been shown, also in other works, that it can conceal some aspects of the transient dynamics or even alter the long-term dynamics, and thus the qualitative behavior, of the original system


total Quasi-Steady-State Approximation

A possible way to overcome the limitations of QSSA in enzyme-catalyzed reactions has been proposed in

They simply proposed that, for conditions when ET and S0 have comparable values, the proper intermediate timescale variable is

This yields

JAM Borghans, RJ De Boer, LA Segel, Extending the Quasi-Steady-State Approximation by Changing Variables, Bull. Math. Biol. 58(1), 43–63, 1996

S(t) = S(t) + C(t)


Validity of tQSSA

Tzafriri and Edelman (J. Theor. Biol., 2004) derived sufficient conditions for the validity of tQSSA, which can be summarized by

that is, the dissociation rate of the enzyme–substrate complex is much faster than the catalytic conversion of substrate into product

Thus, the tQSSA is likely to be an excellent approximation for any ratio of enzyme to substrate and for any ratio of timescales

An interesting reading for applications of tQSSA to several kinds of PINs is

k−1 À k2

A Ciliberto, F Capuani, JJ Tyson, Modeling Networks of Coupled Enzymatic Reactions Using the total Quasi–Steady State Approximation, PLOS Computational Biology 3(3), 463–472, 2007


Outline





Allosteric reaction


Stochastic models




Cooperative Reactions

In the basic model of enzymatic reaction we have assumed that each molecule of enzyme binds only one substrate molecule

It is quite common to have multiple binding sites, e.g. the hemoglobin has four binding sites for oxygen

An enzymatic reaction is said to be cooperative if the binding of one molecule to one site affects the binding affinity at other sites


Allosteric Effect

The mechanism causing such phenomenon is named allosteric effect

A substrate can be an activator or inhibitor, depending on whether it increases or decreases the binding affinity at other sites

If the substrate and the modulator are the same species, then the interaction is called homotropic, otherwise heterotropic


Cooperative Reaction – Example

The simplest cooperative mechanism is shown below

By applying the law of mass action and then the QSSA, analogously to what done for the M–M model it is possible to derive the maximum reaction rate

( ) 200

04200

000 ssKKK

skKkse

dtdssR

mmm

m

t +′+′+′

===

3

34

1

21 ,k

kkK

kkkK mm

−− +=′+

=


Hill Plot

Plotting the reaction rate as a function of s0 it is possible to observe the difference with respect to standard reactions

For the sake of simplicity, we plot the case when k2=0, which yields

2000 0 sRs ∝⇒→

( ) 0,0

000 >

+= n

KsQs

sRm

n

n

In this case the behavior is usually described by a Hill curve


Hill Coefficient

The quantity n is termed the Hill coefficient

The same term can be found by considering an ideal reaction, with an enzyme binding n substrate molecules at the same time (complete cooperativity)

In real cases, the value of n has not to be an integer, rather it is a real number, because usually the substrates do not bind contemporaneously to different sites, thus

n>1 positive cooperation

n<1 negative cooperation

n=1 no cooperation

Therefore the Hill coefficient is a measure of the cooperativity of the reaction


Outline





Allosteric reaction


Stochastic models




Lineweaver–Burk Plot

The L–B plot (or double reciprocal is a common tool in biochemistry analysis of enzyme kinetics

It is easily derived by inverting the equation of S in the M–M model

[ ][ ]SV

SKV

m

max0

1 +=

[ ] maxmax0

11VSV

KV

m +=

It is very useful because it enables linear regression of experimental data


Regulation of Enzymatic Reactions

Enzymes can increase the rate of a reaction by several orders of magnitude, but they can also be used for fine regulation of reaction pathways

The production and degradation are often adapted to the current requirements of the cell

Furthermore, the may be targets of effectors, both inhibitors and activators

The effectors are proteins or other molecules that can modify the enzymatic reaction rate


Enzyme Inhibition

Different types of inhibition mechanism exist, depending on

the state in which the enzyme can bind the effector

the ability of different complexes to release the product

The general pattern of inhibition is shown in the figure below; the several subcases derive by eliminating some of the reactions


Competitive Inhibition

The inhibitor competes with the substrate for the binding site (or inhibits substrate binding by binding elsewhere to the enzyme)

[ ] [ ][ ]SKSV

dtPd

m +=α

max

[ ] [ ][ ][ ]EI

IEKKI

II

=+= ,1α


Competitive Inhibition L–B Plot

The competitive inhibition can be characterized experimentally by means of the L–B plot


Uncompetitive Inhibition

The inhibitor binds only to the ES complex

This may be due to conformational changes of the enzyme caused by the substrate binding (allosteric effect), which makes a new binding site accessible

[ ] [ ][ ]SK

SVdtPd

m α′+= max

[ ] [ ][ ][ ]ESI

IESKKI

II

=′′

+=′ ,1α


Uncompetitive Inhibition L–B Plot

The L–B significantly differs from the previous case


Mixed Inhibition

The inhibitor can bind both to the enzyme and to the ES complex

Therefore it binds to a different site than that used by the substrate

[ ] [ ][ ]SK

SVdtPd

m αα ′+= max

[ ] [ ][ ][ ]ESI

IESKKI

II

=′′

+=′ ,1α

[ ] [ ][ ][ ]EI

IEKKI

II

=+= ,1α


Mixed Inhibition L–B Plot

Also in this case the L – B plot can be used to distinguish from the other types of inhibition mechanism


Noncompetitive and Partial Inhibition

Noncompetitive inhibition can be viewed as a special case of mixed inhibition, when the parameters α and α’ are equal

In this case the inhibitor has the same affinity to the enzyme with or without bound substrate

It is rarely encountered in experimental practice

If the product can also be formed from the enzyme–substrate–inhibitor complex, the inhibition is said to be partial

If the production rate from ESI complex is high, the effect of this latter mechanism can even yield an activating instead of an inhibiting effect


Dependence on Environmental Factors

It is worth pointing out that enzymatic reactions are strongly dependent on factors like pH and temperature, as shown in the two examples below


Outline





Allosteric reaction


Stochastic models




Stochastic Models

The models considered so far are deterministic, i.e. given the initial conditions (the state at t0) and the exogenous signals perturbing the model, the response is univocally determined

A stochastic model, on the contrary, involves random variables, therefore its behavior cannot be predicted a priori, although it can be statistically characterized

The figure shows two realization of the same stochastic process, starting from the same initial condition


Biological Systems are Stochastic

Certainly biological ones fall in the category of stochastic systems, indeed the very basic steps of every molecular reaction can be described only in terms of its probability of occurrence

Moreover, the diffusion of molecules is a realization of a random walk process (Brownian motion)

So, why deterministic model are so widespread?

Usually the phenomena under consideration involve a large number of molecules, therefore the average effect is well described through deterministic equations

When is it necessary to use stochastic models/simulations?

When the mechanism to be described is based on the interaction of few molecules, or we want to simulate the functioning of a little pool of cells


Gillespie Algorithm

The algorithm has been first presented in

The purpose is to simulate chemical reactions with limited computational resources

The outcome is a realization of the underlying stochastic process

DT Gillespie, A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions, J. of Computational Physics 22, 403–434, 1976


Basic Principles

The basic working principles of the Gillespie algorithm derive from the description of the collision of particles in a vessel

When two molecules collide, the reaction happens only if they have proper orientation and kinetic energy

A first hypothesis is that the frequency of non–reacting collisions is much larger than that of reacting ones

Secondly, the algorithm assumes that each reaction involves no more than two molecules


Sketch of the Derivation

The method starts from the observation of the volume occupied by a molecule A that is in relative motion with respect to another molecule B


Sketch of the Derivation

Assuming a random uniform distribution of the molecules in the volume V, the probability of collision can be computed as

If in such volume we have X1 molecules of the species S1 and X2 of the species S2, the probability of collision is

tvrVVV δπδ 122

121

coll−= Probability of collision of two

molecules in the interval �t

tvrVXX δπ 122

121

21−


Stochastic Reaction Constant

In this framework, the reactions can be characterized by means of a probability of reaction per unit time, instead of a kinetic rate as in ODE models

For instance, given the reaction

we can define a constant c1, dependent on the chemo–physical properties of the molecules and on the temperature, such that

X1 X2 c1dt = Probability that the reaction happens in the volume Vwithin the interval dt

In general, given a system of N molecules and M reactions, each reaction is characterized by a specific stochastic reaction constant cµ (µ=1,…,M)

1211 2: SSSR →+


Stochastic and Deterministic Constants

Intuitively, the stochastic reaction constant, cµ, must be linked to the kinetic constant of the corresponding determinist equation, kµ

In the validity ranges of the deterministic models, it is possible to establish the simple relation

The presence of the factor V depends on the fact that deterministic models take as state variables the concentrations, whereas stochastic ones use the number of molecules

11 Vck =


Stochastic Simulation

There are two main approaches to solve the stochastic system and compute the system evolution

Master Equation Approach

Stochastic Simulation Algorithm (SSA) with Gillespie method


Master Equation Approach

The key element in this approach is the probability function

The master equation describes the evolution of the function P

The probability P(X1,…,XN;t+dt) can be derived as a combination of the probability of all the possible reactions that can happen within dt

The differential equation that is derived by this argument does not usually have analytical solution, nor it admits a computationally efficient solution

( )tXXXP N ;,,, 21 KProbability that at time t there are X1molecules of species S1, …, XN of species SN


Outline





Allosteric reaction


Stochastic models




Stochastic Simulation Algorithm

Disregarding the formalism of the master equation, looking for a more practical approach, we need to answer two questions

Starting at time t, which is the next occurring reaction

When this reaction will occur

Clearly, the answer can be given only in terms of probability, thus let us define the pdf of the reaction, P(τ,µ), such that

( ) τµτ dP , Probability that, given X1,…, XN at time t, the next reaction in V will be Rµ and that it will occur within (t+ τ,t+ τ +dτ)



In order to find an analytical expression for P(τ,µ) let define

hµ → number of possible distinct combinations of reactants in Rµ in the state (X1,…,XN), µ=1,…,M

If Rµ has two–reactants (of the type S1+S2 → …), then hµ =X1X2

If Rµ has one reactant (of the type 2S1 → …), then hµ =1/2 X1(X1-1)

aµdt = hµcµdt → probability that a reaction Rµ occur in the interval (t,t+dt) starting from the state (X1,…,XN), µ=1,…,M at time t



From this quantities it is possible to derive the following expression

where

Having a random number generator with uniform distribution, it is possible to generate the exponential distribution above

( ) ( )⎩⎨⎧ =∞≤≤−

=altrimenti

,,100

exp, 0 Mandaa

PKµττ

µτ µ

( )

∑∑==

==

==MM

chaa

Mcha

110

,,1

ννν

νν

µµµ µ K


Steps of the Gillespie Algorithm

Step 0 (Initialization) – Define the number of molecules of each species, the kinetic constants and the random number generator

Step 1 (Monte Carlo) – Generate random numbers, to determine which is the next reaction and the length of the interval dt

Step 2 (Update) – Increase time by dt and update the number of molecules of each species on the basis of the occurred reaction

Step 3 (Iterate) – If the number of reactants is greater than zero and the simulation stop time has not yet been reached, iterate from Step 1


Drawbacks of the Stochastic

High computational load

Not possible to derive reduced order model (like M–M or Hill terms)


References

DL Nelson, MM Cox, Lehninger Principles of Biochemistry, WH Freeman, 2004

JD Murray, Mathematical Biology, Springer, 2007

E.H. Flach, S. Schnell, Use and abuse of the quasi-steady-state approximation, IEE Proc.–Syst. Biol. 153(4), 187–191, 2006

A Ciliberto, F Capuani, JJ Tyson, Modeling Networks of Coupled Enzymatic Reactions Using the totalQuasi–Steady State Approximation, PLOS Computational Biology 3(3), 463–472, 2007

DT Gillespie, A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions, J. of Computational Physics 22, 403–434, 1976

Date post:	23-Jun-2020
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

Modeling of Biochemical Reactionsusers.ece.cmu.edu/~brunos/Lecture2.pdf · Modeling of Biochemical...

Documents