Flexible polymer chain dynamics

Yong-Gu Lee

Statistical mechan-ics of chain mole-cules

(p.3)

End-to-End, Radius of gy-ration

1

2 2

, 0

12 2

0

22 2

0

2

: Radius of gyration: root-mean-square distance of the collection of atoms, or groups,

from their common center of gravity

1

1

n

ii

i j i i ji j i i j n

n

i

iji j

r l

s

s n s

s n r

r l

r r l l l l

2 2

1 0

where 2j

ij i i jn i i i j n

r l

l l

Pg. 8

The freely jointed chain

• No constraint on bond angles• No constraint on bond rotations• Bond length can be considered to be on “average”

Average values

2 2

0

2 2

0

22 2

0

2

2

1

i i ji i j n

i ji j n

iji j n

r l

r nl

s n r

l l

l l

2 2

0

2 2

0

2

2

The subscript zero is applied to denote the unperturbed state in recognition of the disregard

of contributions to from interactions of long range involving remote pairs

i ji j n

r nl

r nl

r i

l l

2 2

0

2 2 22 2 2 2

00 1 1 1

For freely jointed chain 0,

The characteristic ratio defined by

C / is unity for all values of n for the freely jointed chain

1 1 1 1 / 2

i j

n

jn

i j n j k

j

i j

r nl

s l n j i l n k l n j j

l l

2 2

0

2/

6 1

n

ns nl

n

The freely rotating chain (The Porod-Kratky Chain)• Yes constraint on bond angles• No constraint on bond rotations• Bond length can be considered to be on “average”

2 2

0

2 2

0

2

when n , 2

A related quantity of interest is the persistence length,

defined as the average sum of the projections of all bonds j on bond i in an

indefinitely long chai

i ji j n

r nl

r nla nl

i

l l

n. The bond i is taken to be remote from either end of the

chain.

Kuhn invested this manifestly artificial model with a semblance of reality by pointing out that the

correlation between bonds i and i+k in any real chain endowed with a finite degree of flexibility

must vanish as k increases. In random macromolecules of the usual arge n, the correlation will vanish for

k << n. The sum over j of terms for a given value of i converges, for unperturbed chains,

and

i jl l

0

the value of this sum is independent of i, provided that i is well removed from both ends of the chain.

The double sum may therefore be taken to the proportional to n for sufficiently largi ji j n

l l

2

0

e n.

Asymptotic proportionality of to n is thus assured for any flexible, real chain. Unlike the freely jointed chain,

however, the constant of porportionality will depart considerably from unity i

r

n general.

A real chain of sufficient length may be presented, theefore, by an equivalent chain comprising n hyupotetical bonds each of

length l connected by free joints as above. Arbitrariness in th

max

2 2

0

2max 0

e choices of n and l is removed by requiring

that

where is the fully extended length of the real chain and is its actual unperturbed mean swuare

end-to-end length. Thus for a

n l r

n l r

r r

2 2max 0

2 2

0

2 2

0

22 2

0

polymethylene chain with n sufficiently large

0.831 and r 6.7 . It follows that n/n 10 real bonds per equivalent segment.

2

2

1

i i ji i j n

i ji j n

iji j n

r nl nl

r l

r nl

s n r

l l

l l

Bead-spring chain model

A double-stranded DNA molecule is described by a bead-spring chain model composed of Nb beads of hydrodynamic radius a = 77 nm connected by Ns = Nb − 1 entropic springs. Each bead represents a DNA segment of 4850 base pairs, i.e., Nb = 11 corresponds to a stained -DNA, which has a contour length of 22 μm and radius of gyration of 730 nm. The springs connecting the beads obey a worm-like chain force law. Here, bk is the Kuhn length for DNA and Nk,s = 20 is the number of Kuhn length per spring.

Pg 29

Pg 5

Yu Zhang, “Brownian dynamics simulation of DNA in complex geometries,” PhD thesis, University of Wisconsin – Madison, 2011

Increased from natural 16.3um by TOTO-1 dye

Diffusion coefficient

• The projection of the Brownian motion in the x-y plane was tracked using video fluorescent mi-croscopy and the increase in the mean square dis-placments with time t were recorded for an en-semble of molecule paths.

2 2 2x y Dt

Douglas E. Smith, Thomas T. Perkins, and Steven Chu, “Dynamical scaling of DNA diffusion coefficents,” Macromolecules, 1996, pp. 1372-1373

Zimm model for the radius of Gyration

>> 22^(3/5)

ans =

6.3893

Persistent length

• Informally, for pieces of the polymer that are shorter than the persis-tence length, the molecule behaves rather like a flexible elastic rod, while for pieces of the polymer that are much longer than the persis-tence length, the properties can only be described statistically, like a three-dimensional random walk.

• Formally, the persistence length, P, is defined as the length over which correlations in the direction of the tangent are lost. In a more chemi-cal based manner it can also be defined as the average sum of the projections of all bonds j ≥ i on bond i in an indefinitely long chain.

• Let us define the angle θ between a vector that is tangent to the polymer at position 0 (zero) and a tangent vector at a distance L away from position 0, along the contour of the chain. It can be shown that the expectation value of the cosine of the angle falls off exponentially with distance

cospLe

http://en.wikipedia.org/wiki/Persistence_length

Persistent length• For a chain possessing some degree of stiffness, the

tangents to two segments of the chain contour will tend to be pointed in the same direction, provided that the segments are sufficiently close to each other, In other words, the local contour tends to persist in a given direction (segment directions are correlated). If the chain contour is represented by a virtual chain comprising a string of segment vectors, each of length b, the stiffness of the chain can be represented in a quantitative fashion as the sum , of the average pro-jections of each segment vectors on the first seg-ment vector.

1

cosn

i

ni

P b

𝑃𝑛

Paul J. Haggerman, “Flexibility of DNA,” Annual Review of Biophysics, Vol 17. pp. 265-286, 1988

Radius of gyration

• It is defined as the root-mean-square distance of the collection of atoms, or groups, from their com-mon center of gravity.

12 2

0

1

is the distance of atom i from the center of gravity of the chain in a specified configuration.

n

ii

i

s n s

s

Paul J. Flory, “Statistical mechanics of chain molecules,” pp. 4, Hanser Publishers, 1988

Kuhn length

• The Kuhn length is a theoretical treatment, developed by Werner Kuhn, in which a real polymer chain is considered as a collection of N Kuhn segments each with a Kuhn length b. Each Kuhn segment can be thought of as if they are freely jointed with each other. Each segment in a freely jointed chain can randomly orient in any direction without the influence of any forces, independent of the directions taken by other segments. Instead of considering a real chain consisting of n bonds and with fixed bond angles, torsion angles, and bond lengths, Kuhn considered an equivalent ideal chain with N connected segments, now called Kuhn segments, that can orient in any random direc-tion.

Kramers freely jointed bead-rod chain model

P. M. Saville and E. M. Sevick, “collision of a field-driven polymer with a finite-sized obstacle: ABrownian dynamics simulation,” Macromolecules, Vol. 32, pp. 892-899, 1999

Length based sortingP. M. Saville and E. M. Sevick, “collision of a field-driven polymer with a finite-sized obstacle: ABrownian dynamics simulation,” Macromolecules, Vol. 32, pp. 892-899, 1999

Experimental parameters for Kramers freely joined bead-rod chain model

• As in previous work, fluorescently tagged Lambda-DNA is modeled using the Kramers freely jointed bead-rod chain model. This model has demonstrated quanti-tative agreement with experiments in single molecule flow studies of DNA even though the model does not reproduce the correct bending forces or effective en-tropic spring force of the wormlike chain or Kratky-Porod model. To simulate the Kratky-Porod model, however, in the present application, 10 sub-Kuhn step level dis-cretization points are needed in order to produce the correct persistence length in a discrete model appropriate for Brownian dynamics. Thus, the number of dynamics variables becomes prohibitively large for the long time, large ensemble simulations considered in this work (see Ensemble Size section below). Therefore, we choose to focus on the qualitative comparisons to the experimental data with our simulations. As we shall discuss below, errors associated with ensemble size in these comparisons are at least as important as those associated with the shortcomings of the model. In the Kramers model, the polyelectrolyte is divided into N beads, at which the mass and hydrodynamic drag are concentrated, connected by N - 1 massless rods. The en-tire molecule has a contour length of 21 um and is modeled using 150 beads, yield-ing a Kuhn step size, bk, of 0.142 um. The drag is assumed to be isotropic, and hy-drodynamic interactions (HI) between segments of the chain are ignored.Nerayo P. Teclemariam, Victor A. Beck, Eric S. G. Shaqfeh,,§ and Susan J. Muller,

“Dynamics of DNA Polymers in Post Arrays: Comparison of Single Molecule Experiments and Simulations,” Macromolecules, Vol. 40, pp. 3848-3859, 2007

>> 150* 0.142

ans =

21.3000

Polymer chain dynamics

The Brownian dynamics algorithm is implemented by integrating the stochastic differential equations to first order in t

For a time step t the displacement of bead consists of two parts:

tt t

r r

t .

The first term t is the position which would have been reached in the absence of constratints. The second term

is the correction to t due to the constraint forces.

The bead positi

t

t

r

r

r r

on after an unconstrained move, t ,

is calculated in a straightforward manner:

t t t

: transpose of the velocity gradient tensor

= 2

2 t

is

B

i i ii

t

t t t

k T

a B t

r

Fr r κ r W

κ

r u

W

chosen from a Gaussian distribtuion characterized by the first and second moments:

0,

t.

is the rod like link length.

t

t t

a

W

W W δ

Tony W. Liu. “Flexible polymer chain dynamics and rheological properties in steady flows,” Journal of Chemical Physics, Vol 90. No. 5826, 1989

• /* Rouse matrix *** Aij*/• MatrixNminus1d* getRouseMatrix()• /* Modified Rouse matrix *** AHatij*/• MatrixNminus1d* getModifiedRouseMatrix(Matrix3Nminus1d& umatrix)• /* Kramers matrix *** Cij*/• MatrixNminus1d* getKramersMatrix()• /* Kramers tensors matrix is second rank tensor *** Kij*/• Matrix3d* getKramersTensorMatrix(int j, int k, Matrix3Nminus1d& umatrix)• /* B_bar_ij*/• MatrixNminus1_Nd* getB_BarMatrix()• const double physical_coefficient = 1.0 / a * sqrt(2.0 * boltzman_constant *

temperature/friction_coefficient);• Matrix3d kappa;//Transpose of the velocity gadient tensor del(v)

int _tmain(int argc, _TCHAR* argv[]){

Matrix3d kappa;//Transpose of the velocity gadient tensor del(v)kappa << 0,0,0, 0,0,0, 0,0,0;const double maxt = 100;const double deltat = double(1)/double(maxt)*1000.0e-7;//1000e-12;

////The whole simulation duration is 1000 pico secondconst double boltzman_constant=1.3806503e-23;

//1.3806503 × 10-23 m2 kg s-2 K-1const double temperature = 298.0;const double friction_coefficient = 6.0 * M_PI * 8.9e-4 * 10.0e-9;

// 6 PI Mew R, 10 nm radius bead // The dynamic viscosity of water is 8.90 × 10ʬ4 Pa·s at about 25 °C.const double a = 100.0e-9; // rod lengthconst double alpha = sqrt(2.0 * boltzman_constant * temperature *

friction_coefficient);const double physical_coefficient = 1.0 / a * sqrt(2.0 * boltzman_constant

* temperature/friction_coefficient);

VectorNminus1d ux, uy, uz;for( int i = 0; i < N-1; ++ i ){

ux(i) = 0;uy(i) = -1;uz(i) = 0;

}

Matrix3Nminus1d umatrix;Matrix3Nminus1d u_primematrix;Matrix3Nd rmatrix;Matrix3Nd r_primematrix;Matrix3Nd compensating_matrix;MatrixNminus1d* RouseMatrix;RouseMatrix = getRouseMatrix();MatrixNminus1_Nd* B_Bar_Matrix;B_Bar_Matrix = getB_BarMatrix();MatrixN_Nminus1d* B_Matrix;B_Matrix = getB_Matrix();

for( int m = 0; m < N-1; ++m ){

umatrix(0,m) = ux(m);umatrix(1,m) = uy(m);umatrix(2,m) = uz(m);

}rmatrix = (a*(*B_Matrix) * umatrix.transpose()).transpose();

// 3 x N//std::cout << rmatrix << std::endl;

//print the initial coordinates of the beadsofstream file;file.open("E:/temp/beads.xyz"); //open a file

double mu = 0.0; double sigma = 1.0;

std::tr1::normal_distribution<double> normal(mu, sigma);for( int t = 0; t < maxt; ++t){

for( int m = 0; m < N; ++m){

_Vector3d *delta_w;delta_w = get_delta_w(normal, deltat);r_primematrix.col(m) = rmatrix.col(m) + kappa *

rmatrix.col(m) * deltat + alpha / friction_coefficient * (*delta_w);delete delta_w;

}//m//std::cout << r_primematrix << std::endl; //print the

unconstrained movement of the beads (Eq. 65)for( int m = 0; m < N-1; ++m){

u_primematrix.col(m) = (r_primematrix.col(m+1)-r_primematrix.col(m))/a;

umatrix.col(m) = (rmatrix.col(m+1)-rmatrix.col(m))/a;}VectorNminus1d gamma;get_gamma(gamma, umatrix, u_primematrix, deltat); //Eq. 70for( int n = 0; n < N; ++n ){

rmatrix.col(n) = r_primematrix.col(n);for( int m = 0; m < N-1; ++m){

rmatrix.col(n) += -2.0 * deltat / friction_coefficient * a * gamma(m)*(*B_Bar_Matrix).coeff(m,n)*umatrix.col(m);

}}//std::cout << "t = " << t << "-------------" << std::endl;_Vector3d babo;file << N << "\n";//No. of filesfile << "beads.xyz time [ps] " << float(t) << "\n";//Time

stampfor( int m = 0; m < N; ++m){

//std::cout << "\tm = " << m << std::endl;//std::cout << "\t" << rmatrix.col(m).coeff(0) << ' ' <<

rmatrix.col(m).coeff(1) << ' ' << rmatrix.col(m).coeff(2) << ' ' << std::endl;

file << "beads " << rmatrix.col(m).coeff(0) << ' ' << rmatrix.col(m).coeff(1) << ' ' << rmatrix.col(m).coeff(2) << ' ' << "\n";

}

}//tfile.close();

return 0;}

std::tr1::normal_distribution<double> normal(mu, sigma);for( int t = 0; t < maxt; ++t){

for( int m = 0; m < N; ++m){

_Vector3d *delta_w;delta_w = get_delta_w(normal, deltat);r_primematrix.col(m) = rmatrix.col(m) + kappa *

rmatrix.col(m) * deltat + alpha / friction_coefficient * (*delta_w);delete delta_w;

}//m//std::cout << r_primematrix << std::endl; //print the

unconstrained movement of the beads (Eq. 65)for( int m = 0; m < N-1; ++m){

u_primematrix.col(m) = (r_primematrix.col(m+1)-r_primematrix.col(m))/a;

umatrix.col(m) = (rmatrix.col(m+1)-rmatrix.col(m))/a;}VectorNminus1d gamma;get_gamma(gamma, umatrix, u_primematrix, deltat); //Eq. 70for( int n = 0; n < N; ++n ){

rmatrix.col(n) = r_primematrix.col(n);for( int m = 0; m < N-1; ++m){

rmatrix.col(n) += -2.0 * deltat / friction_coefficient * a * gamma(m)*(*B_Bar_Matrix).coeff(m,n)*umatrix.col(m);

}}//std::cout << "t = " << t << "-------------" << std::endl;_Vector3d babo;file << N << "\n";//No. of filesfile << "beads.xyz time [ps] " << float(t) << "\n";//Time

stampfor( int m = 0; m < N; ++m){

//std::cout << "\tm = " << m << std::endl;//std::cout << "\t" << rmatrix.col(m).coeff(0) << ' ' <<

rmatrix.col(m).coeff(1) << ' ' << rmatrix.col(m).coeff(2) << ' ' << std::endl;

file << "beads " << rmatrix.col(m).coeff(0) << ' ' << rmatrix.col(m).coeff(1) << ' ' << rmatrix.col(m).coeff(2) << ' ' << "\n";

}

}//tfile.close();

return 0;}

std::tr1::normal_distribution<double> normal(mu, sigma);for( int t = 0; t < maxt; ++t){

for( int m = 0; m < N; ++m){

_Vector3d *delta_w;delta_w = get_delta_w(normal, deltat);r_primematrix.col(m) = rmatrix.col(m) + kappa *

rmatrix.col(m) * deltat + alpha / friction_coefficient * (*delta_w);delete delta_w;

}//m//std::cout << r_primematrix << std::endl; //print the

unconstrained movement of the beads (Eq. 65)for( int m = 0; m < N-1; ++m){

u_primematrix.col(m) = (r_primematrix.col(m+1)-r_primematrix.col(m))/a;

umatrix.col(m) = (rmatrix.col(m+1)-rmatrix.col(m))/a;}VectorNminus1d gamma;get_gamma(gamma, umatrix, u_primematrix, deltat); //Eq. 70for( int n = 0; n < N; ++n ){

rmatrix.col(n) = r_primematrix.col(n);for( int m = 0; m < N-1; ++m){

rmatrix.col(n) += -2.0 * deltat / friction_coefficient * a * gamma(m)*(*B_Bar_Matrix).coeff(m,n)*umatrix.col(m);

}}//std::cout << "t = " << t << "-------------" << std::endl;_Vector3d babo;file << N << "\n";//No. of filesfile << "beads.xyz time [ps] " << float(t) << "\n";//Time

stampfor( int m = 0; m < N; ++m){

//std::cout << "\tm = " << m << std::endl;//std::cout << "\t" << rmatrix.col(m).coeff(0) << ' ' <<

rmatrix.col(m).coeff(1) << ' ' << rmatrix.col(m).coeff(2) << ' ' << std::endl;

file << "beads " << rmatrix.col(m).coeff(0) << ' ' << rmatrix.col(m).coeff(1) << ' ' << rmatrix.col(m).coeff(2) << ' ' << "\n";

}

}//tfile.close();

return 0;}