Ring closure in actin polymers

TL;DR

This paper analyzes the probability of ring closure in actin polymers modeled as semiflexible Worm Like Chains, providing predictions that can be tested experimentally and discussing implications for bend angle fluctuations.

Contribution

It offers a theoretical analysis of ring closure probability in actin polymers within the WLC model, linking it to experimental testing and fluctuation effects.

Findings

Predicted ring closure probability for actin polymers

Analysis of bend angle fluctuations due to ring closure

Potential experimental validation with fluorescence experiments

Abstract

We present an analysis for the ring closure probability of semiflexible polymers within the pure bend Worm Like Chain (WLC) model. The ring closure probability predicted from our analysis can be tested against fluorescent actin cyclization experiments.We also discuss the effect of ring closure on bend angle fluctuations in actin polymers.

Equations

$$\mathcal{E}[\mathcal{C}]=\frac{A}{2}{\int }_0^{L}ds{\kappa }^2$$

$$\tilde{Q}(\vec{r})$$

$$P(z)=\int d\vec{r}\tilde{Q}(\vec{r})\delta (r_3-z),$$

$$\tilde{P}(f)={\int }_{-L}^{L}P(z)e^{\frac{fz}{L_P}}dz$$

$$\int Q(\vec{\rho })d\vec{\rho }$$

$$\int \frac{Q(\vec{\rho })}{{\beta }^3}d\vec{r}$$

$$\frac{Q(0)}{{\beta }^3}$$

$$<{\varphi }^2{>}_{ring}^{xy}=\frac{1}{12}(1-\frac{6}{{\pi }^2})\frac{L}{L_P}$$

$${\int }_0^L{\varphi }_z(s)ds=0.$$

$$<{\varphi }^2{>}_{ring}^z=\frac{1}{12}(1-\frac{6}{{\pi }^2})\frac{L}{L_P}$$

$$<{\varphi }^2{>}_{ring}^{3d}=\frac{1}{6}(1-\frac{6}{{\pi }^2})\frac{L}{L_P}$$

$$<{\varphi }^2{>}_{lin}^{3d}=\frac{1}{3}\frac{L}{L_P}$$

$$\tilde{Q}(\vec{r})$$

$$P(z)=\int d\vec{r}\tilde{Q}(\vec{r})\delta (r_3-z),$$

$$\tilde{P}(f)={\int }_{-L}^{L}P(z)e^{\frac{fz}{L_P}}dz$$

$$Z(f)=\mathcal{N}\int \mathcal{D}\left[\hat{t}(s)\right]\exp \left\{-\frac{L_P}{2}\left[{\int }_o^L{\left(\frac{d\hat{t}}{ds}\right)}^{2}ds\right]\right\}$$

$$\times \exp \left[\frac{f}{L_P}{\int }_0^L{\hat{t}}_{2}ds\right]$$

$$Z(f)=\mathcal{N}\int \mathcal{D}\left[\hat{t}(\tau )\right]\exp \left\{-{\int }_o^{\beta }d\tau \left[\frac{1}{2}{\left(\frac{d\hat{t}}{d\tau }\right)}^2-f{\hat{t}}_z\right]\right\}$$

$$Z(f)=\sum _n{e}^{-[\beta E_n]}{\psi }_n\left({\hat{t}}_A\right){\psi }_n\left({\hat{t}}_{\beta }\right)$$

$$Z(f)=⟨o\left|\exp -\beta H_f\right|o⟩$$

$$H_{ll}=\frac{l(l+1)}{2}$$

$$H_{ll+1}=f(l+1)\sqrt{1/[(2l+1)(2l+3)}]$$

Full text

I I. Introduction

In the past two decades, there has been much interest in the theoretical study of semiflexible polymer elasticity. These studies are motivated by micromanipulation experimentsBustamante et al. (2000); Forth et al. (2008); Daniels et al. (2009) on biopolymers. In particular, in recent years there have been experiments involving stretching DNA moleculesBustamante et al. (2000) which give us information about the bend elastic properties of DNA. There have also been experiments on fluorescently tagged actin filamentsOtt et al. (1993) where they measure the bend persistence length of actin. More recently, there have been fluorescence experiments on cyclization of actin filamentsSanchez et al. (2010). In these papers they analyze the formation of rings in actin polymers and study the effect of ring closure on bend angle fluctuations in these polymeric rings. Our interest here is limited to the process of cyclization itself and therefore in our analysis we restrict to polymers with only bend degrees of freedom and no twist degree of freedom. Actin cyclization is of interest to biologists Mavrakis et al. (2014) who do visualization studies of actin ring formation in the context of cell division.

II **II. **Ring Closure Probability Distribution

Our starting point is the pure bend Worm Like Chain (WLC) modelSamuel and Sinha (2002). In this model, the polymer configuration is viewed as a space curve ${\vec{x}(s)}$. There is a tangent vector associated with each point on the polymer of contour length $L$ and the energy of configuration is given by:

[TABLE]

where $\cal{C}$ stands for the polymer configuration. $A$ is the bending elastic constant and the curvature $\kappa=|\frac{d{\hat{t}}}{ds}|$.

One of the key quantities characterizing the elasticity of a biopolymer is $\tilde{Q}(\vec{r})$, the probability distribution for the end to end distance vector $\vec{r}$ between the two ends of the polymer as it gets jiggled around by thermal fluctuations in a cellular environment Samuel and Sinha (2002). In Samuel and Sinha (2002) we use a method for solving the wormlike chain model for semiflexible polymers to any desired accuracy over the entire range of polymer lengths to determine $\tilde{Q}(\vec{r})$. The plots for $\tilde{Q}(\vec{r})$ for various $\beta=\frac{L}{L_{P}}$, the ratio of the contour length $L$ to the persistence length $L_{P}$, reveal the dependence of the end to end distance vector on the rigidity of the polymer (See Fig. [4] in Samuel and Sinha (2002)).

We outline the theoretical calculation of $\tilde{Q}({\vec{r}})$ below. (For a detailed exposition please see Appendix $A$). Consider a situation where the initial and final tangent vectors (${\hat{t}}_{A}=\frac{d{\vec{x}}}{ds}|_{s=0}$ and ${\hat{t}}_{B}=\frac{d{\vec{x}}}{ds}|_{s=L}$) are held fixed. Then $\tilde{Q}({\vec{r}})$ has the following path integral representation:

[TABLE]

Here ${\cal N}$ is the normalization constant and $k_{B}T$, the thermal energy at temperature $T$. As mentioned in Samuel and Sinha (2002), we solve for $\tilde{Q}(\vec{r})$ by first considering a related end to end distance measure:

[TABLE]

which is $\tilde{Q}(\vec{r})$ integrated over a plane of constant $z$. This in turn is related to ${\tilde{P}}(f)$, the Laplace transform of $P(z)$ given by:

[TABLE]

$f$, the variable conjugate to $z$ has the interpretation of a stretching force and thus ${\tilde{P}}(f)$, can be written as the ratio $Z(f)/Z(0)$ of the partition functions in the presence and absence of an external stretching force $f$. We do an eigenspectrum analysis of ${\tilde{P}}(f)$ and determine $\tilde{Q}(\vec{r})$ using tomographic transformations outlined in Samuel and Sinha (2002).

Here we address a question which is of current interest to application of polymer physics to biology: cyclization of actin filamentsSanchez et al. (2010). Within the pure bend Worm Like Chain (WLC) Model we compute the ring closure probability (RCP) by considering $\tilde{Q}(\vec{r}=\vec{0})$.

III III. Method

In Fig. $4$ of Samuel and Sinha (2002) we display a family of curves of $Q(\rho)$ versus $\rho$, with $\rho=\frac{|{\vec{r}}|}{\beta}$ for various values of $\beta$. $Q(\rho)$ is a theoretically convenient quantity expressed in terms of scaled units ($\vec{\rho}=\frac{{\vec{r}}}{\beta}$). In order to compute the ring closure probability density $\tilde{Q}(\vec{r}=\vec{0})$ we need to change variables from $\rho=\frac{|{\vec{r}}|}{\beta}$ to $|{\vec{r}}|=r$. Setting $\tilde{Q}({\vec{r}})=Q_{\vec{r}}$, we get:

[TABLE]

or

[TABLE]

which in turn implies

[TABLE]

We compute $Q(0)$ for a range of values of $\beta$ using Mathematica. As we can see from the plot of the ring closure probability density $Q(0)$ versus $\beta$ (Fig. $1$), that $Q(0)$ has a small value for short polymers which are hard to bend and form rings and it has a large value for long polymers which are easy to bend and thus the probability density of ring formation is high. We then compute and plot the ring closure probability density in physical space, $Q_{0}=\frac{Q(0)}{\beta^{3}}$ as a function of $\beta$ (Fig. $2$). The qualitative features of the plot shown in Fig. $2$ are in agreement with our intuition. The ring closure probability density $Q_{0}$ in physical space, which is an experimentally measurable quantity is small for very short and long strands of the polymer and peaks around intermediate contour lengths of $L\approx 3L_{P}$ (See Fig. 7-41 on page 438 of Alberts et al. (2008)).

IV IV. Mean squared tangent angle fluctuation

One of the experimentally relevant quantities of interest is the mean squared tangent angle fluctuationSanchez et al. (2010). In Ref. Sanchez et al. (2010) the mean squared tangent angle fluctuation has been calculated for a ring and a linear filament in a two dimensional setup. They find good agreement with experimental measurements.

Here we present a similar calculation in a three dimensional geometry. Consider a polymer configuration in a closed circular ring lying in the $x-y$ plane. Expanding the bend angle fluctuation $\phi(s)$ in a Fourier series and imposing the ring closure constraint and removing zero modes which do not contribute, we find that the contribution from the $x-y$ plane is given by

[TABLE]

We need to add this contribution to the contribution coming from the $z$ direction where the ring closure condition is of the form

[TABLE]

In this case the Fourier expansion for $\phi_{z}(s)$ can be expressed as $\phi_{z}(s)=\sum_{n=2}^{\infty}{\phi_{n}{e^{\frac{2\pi ins}{L}}}}$ which finally gives us

[TABLE]

Thus combining Eqs. 7 and 8, the net mean squared tangent angle fluctuation for a three dimensional ring is given by

[TABLE]

A similar calculation for a linear filament in three dimensions gives us

[TABLE]

We have plotted (9) and (10) in Fig. $3$. These predictions can be tested against future experiments on fluorescently tagged actin filaments. Notice that, as in the two dimensional caseSanchez et al. (2010), we find that $<\phi^{2}>$ is suppressed for a ringlike structure compared to a linear filament. This indicates that cyclization is entropically costly. Also, as expected, the fluctuations are smaller in the two dimensional geometry compared to the three dimensional geometry.

V V. Conclusion

Our treatment is an analysis based on the pure bend Worm Like Chain Model. The absence of twist degree of freedom enables our predictions to be directly tested against actin cyclization experiments where the two ends of the polymer come together without any relative twist between the two ends. This is to be contrasted with analysis of J factor of DNA with twist degree of freedom where the additional twist degree of freedom makes the analysis considerably more cumbersomeSamuel and Sinha (2003); Spakowitz (2006); Shimada and Yamakawa (1984). In Shimada and Yamakawa (1984) an interpolation formula is presented in the intermediate rigidity regime. They Shimada and Yamakawa (1984) also presented analytical expressions for the ring closure probability density $Q_{0}$ in the limit of $\beta<<1$ and $\beta>>1$. However, they did not have an exact expression for the entire range of polymer lengths. In contrast, here we present a semianalytical study which gives an essentially exact prediction for $Q_{0}$ for the entire range of rigidity (See Fig. $1$ and Fig. $2$). It would be interesting to see how our predictions quantitatively compare with future cyclization probability data for actin filaments. We also expect our predictions for the mean squared tangent angle fluctuation to be tested against future experiments on fluorescently tagged ring like and linear actin filaments in a three dimensional geometry.

VI Acknowledgement

One of us (SC) would like to thank RRI for providing hospitality during stay in Raman Research Institute as a Visiting Student.

VII Appendix A: Computation of $Q_{0}$

Our goal is to calculate $Q_{0}$, the ring closure probability. $Q_{0}$ is $\tilde{Q}(\vec{r})$ for $\vec{r}=\vec{0}$. $\tilde{Q}(\vec{r})$, the probability distribution for the end to end vector $\vec{r}$ for a semiflexible polymer has the following path integral representation:

[TABLE]

Here ${\cal N}$ is the normalization constant and $k_{B}T$, the thermal energy at temperature $T$.

Instead of $\tilde{Q}(\vec{r})$ it turns out to be easier to first consider $P(z)$

[TABLE]

which is $\tilde{Q}({\vec{r}})$ integrated over a plane of constant $z$.

$P(z)$ in turn is related to ${\tilde{P}}(f)$, the Laplace transform of $P(z)$ given by:

[TABLE]

$f$, the variable conjugate to $z$ has the interpretation of a stretching force and thus ${\tilde{P}}(f)$, can be written as the ratio $Z(f)/Z(0)$ of the partition functions in the presence and absence of an external stretching force $f$.

$Z(f)$ is given by

[TABLE]

which in turn can be expressed as

[TABLE]

which has the interpretation of the kernel of a quantum particle on the surface of a sphere at an inverse temperature $\beta$. We now exploit the analogy between time imaginary quantum mechanics and classical statistical mechanics to re-express $Z(f)$ as follows:

[TABLE]

where $\{\psi_{n}\left(\hat{t}_{A}\right)\}$ is a complete set of normalized eigenstates of the Hamiltonian $H_{f}=-\frac{\nabla^{2}}{2}-f\cos\theta$ and $E_{n}$ are the corresponding eigenstates. For free boundary conditions for the end tangent vectors we can express $Z(f)$ as

[TABLE]

The Hamiltonian $H_{f}=-\frac{\nabla^{2}}{2}-f\cos\theta$ is the Hamiltonian of a rigid rotor in a potential and $|0>$ is the ground state of the free Hamiltonian $H_{0}=-\frac{\nabla^{2}}{2}$. We numerically evaluate $Z(f)$ by choosing a suitable basis in which $H_{f}$ has a tridiagonal symmetric matrix structure with

[TABLE]

[TABLE]

To summarize, after casting the problem analytically we use Mathematica programs to sequentially compute $Z(f)$ and ${\tilde{P}}(f)$, then $P(z)$, then $S(r)=-2rdP(r)/dr=4\pi r^{2}\tilde{Q}(r)$ and finally $\tilde{Q}(\vec{r})$. We then consider the scaled variable $\vec{\rho}=\frac{\vec{r}}{\beta}$. $Q(0)$ is then computed by considering $Q(\vec{\rho})$ at $\vec{\rho}=\vec{0}$ and plotting it as a function of $\beta$. $Q_{0}=\frac{Q(0)}{\beta^{3}}$. Below we display the programs for computing $Q(\vec{\rho})$ and $Q_{0}$. We have inserted some comments as part of the programs for clarity.

Program for computing $Q(\vec{\rho})$

ClearAll[h,f,Z,lmax,H,beta,L1,L2,LPR,PR]

lmax=10;

Nmax=3000;

beta=3;

h=.005;

L={};

For[$n=0$,$n<Nmax+1$,n++,

f=hnI;

$H=Table[Switch[i-j,-1,f*(i+1)/Sqrt[\\ (2i+1)(2i+3)],0,i(i+1)/2,1,f*(i)/Sqrt[(2i-1)(2i+1)],_{,}0],{i,0,lmax},{j,0,lmax}]$;

M=MatrixExp[-beta*H];

(Computation of Z(f))

Z=M[[1,1]];

L=Append[L,Z]]

L=Re[L];

Pz={};

P1z={}

For[l=-2,l¡1200,l++,

xi=.001*l;

P=(h*beta/Pi)*Sum[L[[n]]Cos[(n-1)hxibeta],n,1,Nmax];

Pz=Append[Pz,xi,P];

P1z=Append[P1z,P]];

V=P1z;

QR1=;

L1=Drop[V,2]

L2=Drop[V,-2]

LPR=(L1-L2)/(.001*2);

LPR=Drop[LPR,1];

(Computation of S(r))

$QR=Table[LPR[[i]]*1/((i-1)*.001)*[-1/(2*Pi)],{i,2,1199}]$;

(Computation of Q(r))

$QR1=Table[{(i-1)*.001,LPR[[i]]*(1/(i-1))}*\\ {1/((i-1)*.001)*[-1/(2*Pi)]},{i,2,2}]$;

ListPlot[QR1]

Program for computing $Q_{0}$

ClearAll[h, f, Z, lmax, H, beta, Nmax, L1, L2, LPR, PR]

lmax = 10;

h = .005;

final = {};

For[k = 0, $k<50$, k++,

beta = .25*k + 1;

Nmax = Piecewise[90000, beta $<=3$, 9000, beta $>3$];

L = {};

For[n = 0, $n<Nmax+1$, n++, f = hnI;

$H=Table[Switch[i-j,-1,f*(i+1)/Sqrt[\\ (2i+1)(2i+3)],0,i(i+1)/2,1,f*(i)/Sqrt[(2i-1)(2i+1)],_{,}0],{i,0,lmax},{j,0,lmax}]$;

M = MatrixExp[-beta*H];

Z = M[[1, 1]];

L = Append[L, Z]]

L = Re[L];

Pz = {};

P1z = {};

For[$l=-2,l<1200,l++,xi=.001*l$;

P = (h*beta/Pi)*Sum[L[[n]]Cos[(n - 1)hxibeta], n, 1, Nmax];

Pz = Append[Pz, xi, P];

P1z = Append[P1z, P]];

V = P1z;

QR1 = {};

L1 = Drop[V, 2];

L2 = Drop[V, -2];

LPR = (L1 - L2)/(.001*2);

LPR = Drop[LPR, 1];

$QR=Table[LPR[[i]]*1/((i-1)*.001)*[-1/(2*Pi)],{i,2,1199}]$;

$QR1=Table[{(i-1)*.001,LPR[[i]]}*\\ {(1/(i-1))*1/((i-1)*.001)*[-1/(2*Pi)]},{i,2,2}]$;

$final=Append[final,{beta,(1/beta^{3})*QR[[1]]}]$]

ListPlot[final]