Theoretical Investigations On A Few Biomolecular Rate Processes
Abstract
Traditional topics such as physics, chemistry and mathematics have immensely changed the world in the twentieth century, but the twentyfirst century seems to be that of soft condensed matter physics, which has already shown its tremendous possibilities to influence the everyday human life through its technological manifestations such as biotechnology and soft nano technology. Unlike the traditional topics, soft condensed matter physics has an interdisciplinary nature. It studies systems that usually come under chemistry or biology, using the methods of physics and mathematics and hence, transcends the frontiers between the subjects. Soft matter may be classified into three main classes; colloidal dispersions, polymers and polymer melts and liquid crystals. Study of single polymer chains is a fascinating topic that provides insights to understand many processes occurring in biological systems. Here, we present analytical studies of a few such processes, involving single polymer chains.
In fact, there are a number of biological processes, which involve the dynamics of a single polymer chain. Due to the importance of Brownian motion at the mesoscopic level, soft matter systems are always studied using the analytical as well as computational methods of statistical mechanics. The statistical mechanics of polymers has been developed into a fascinating topic due to the contributions from the theory of random walks and path integrals. The dynamical behavior of manyparticle systems has been described traditionally by the socalled rate theories. Here, we use these classical approaches to study a few biological processes that involve single polymer chains. The kind of processes that we have investigated may be categorized into two, namely the processes that leads to conformational changes in a chain molecule and the processes involving spatial translocation of a polymer. In the first category, we have considered the dynamics of semiflexible polymer loops. Loop formation of chain molecules has a key role in biological processes like DNA replication, gene regulation and protein folding. Hence, the dynamics of a polymer closing to form a loop as well as opening of the loop are topics of considerable theoretical/experimental interest. For closing, results are available in the completely flexible limit. Wilemski and Fixman, (J. Chem. Phys.60, 878 (1974)) have studied the closing and opening reactions in a single flexible polymer chain and using their approach Doi found the closing time to vary with the length of the chain as L2 . Szabo, Schulten, and Schulten, (J. Chem. Phys. 72, 4350 (1980)) have used mean first passage time approach and they find that the closure time vary as L3/2. Both approaches have been compared with simulations (Pastor et.al, J. Chem. Phys., 105, 3878 (1996), Srinivas et. al,116,7276 (2002)). In the case of semiflexible chains, studies are fewer in comparison. However, real polymers such as DNA, RNA or proteins are not flexible and therefore, it is important to incorporate the intrinsic stiffness of the chain into account. In the wormlike chain model, the chain is described as a continuous, inextensible and differentiable space curve represented by the position vector r(s), where s is the arc length. Inextensibility of the chain requires that the tangent vector, u(s) = ∂r(s)/∂s, at any point on the curve must have unit magnitude, i.e, u(s) = 1. But incorporating this constraint has been a difficult problem in dealing with semiflexible polymers. Yamakawa and Stockmayer (J. Chem. Phys., 57, 2843 (1972)) and Shimada and Yamakawa (Macromolecules, 17, 689 (1984)) have calculated ring closure probabilities for wormlike chains and helical wormlike chains. Cherayil and Dua (J. Chem. Phys., 116, 399 (2002)) have calculated closure time for a semiflexible chain using the approximate model for semiflexible chains by Harnau, Winkler and Reineker (J. Chem. Phys., 101, 8119 (1994)) and find that the closure time ~ Lν where ν is in the range 2.2 to 2.4. Physically, one expects that the closing time should decrease exponentially with length in the very short chain limit and then increase with length for longer chains. Hence, the closing time has a minimum at an intermediate length. The reason for this behavior is that, for short chains, the bending energy contributes significantly to the activation energy for the process. The activation energy ~ const./L and hence, the closing time τ ~ exp(const./L). For longer chains, the free energy barrier for closing is due to the configurational entropy and hence, τ obeys a power law. Recently, Jun et. al (Europhys. Lett., 64, 420 (2003)) have followed an approximate one dimensional Kramers approach to reproduce this behavior and obtained the minimum at a length Lmin = 3.4lp, where lp is the persistence length of the chain. Monte Carlo simulations by Chen et.al (Europhys. Lett., 65, 407 (2004)) lead Lmin = 2.85lp.
We investigate (K. P. Santo and K. L. Sebastian, Phys. Rev. E, 73, 031923, (2006)) in detail the problem of loop opening for semiflexible polymers. The inextensibility constraint is incorporated rigorously by setting u(s) to be a unit vector in the angular direction (θ, φ) and the conformations of the polymer are then represented by Brownian motion over a unit sphere in the tangent vector space. We use the wormlike chain model, which takes into account the bending rigidity of the polymer. The bending energy can then be given in terms of the angle coordinates θ and φ. For the dynamics, we make use of a semiclassical approach, which is based on expanding the bending energy about a minimum energy path. For the sake of simplicity, we take the great circle on the unit sphere to be the minimum energy configuration of the loop and expand the bending energy up to second order in terms of fluctuations about this configuration. We find that, this is a very good approximation in the large stiffness limit, as this approach leads to a minimum energy value, which is very close to the exact calculations.
The loop is unstable, unless the ends are bound to each other with a potential. Once the two ends have been brought together, they can separate from each other in any of the three directions in space. Considering the ring to be in the XY plane with the ends meeting in the Yaxis, we find that the separation in the X and Z directions are unstable as motion in these directions lead to decrease in bending energy. But the motion in the other direction, that is, the Y direction leads to increase in energy and is stable. Therefore, we choose the potential to be of Morse type in the Xdirection and stable harmonic ones in the other two directions. With this, the potential energy surface for opening can be found and the rate of opening can be calculated using classical Transition State Theory.
The effects of friction on the rate can also be incorporated using the standard coupling to a bath of harmonic oscillators . We find that for short chains, the rate is strongly length dependent and is welldescribed by the equation Aexp(B/x)/xν, with A and B constants, x = L/lp, L the length of the chain, lp the persistence length and ν ~ 1.2. However, for long chains, the rate is found to obey a power law. But in view of the fact that our approximations, while sensible for short semiflexible chains, are not expected to be valid for long flexible chains and therefore, this result is not expected to be correct.
We also present results for the seemingly more biologically important reverse process, the closing of a semiflexible polymer, thus presenting a rather complete theory of dynamics of semiflexible polymer loops. In this work, we give a detailed multidimensional analysis of the closing dynamics of semiflexible chains by making use of the approximation scheme developed in the previous study of loop opening. We use the formalism of Wilemski and Fixman for the diffusioncontrolled intrachain reactions of polymers and their "closure" approximation for an arbitrary sink function. In this procedure, the closing time is expressed in terms of a sinksink correlation function. We calculate this sinksink correlation function through a normal mode analysis on the chain. The closing times, τclose for different lengths of the chain are then obtained. We find that τclose(L) ~ L4.5W(L), where W(L) was found to be described by B'exp(A'/L) with A' and B' constants. τclose(L) is found to have a minimum at Lmin = 2.4lp, which is to be compared with the values obtained through a one dimensional analysis (Europhys. Lett., 64, 420 (2003)) and simulations (Europhys. Lett., 65, 407 (2004)). We thus present a multidimensional analysis that give results that are physically expected. There does not seem to be any previous analysis which leads to these results shown through onedimensional studies and simulations.
In the category of translocation problems, we consider DNA packaging in viruses. DNA Packaging into the viral capsid is an essential step in any kind of viral infection. The mechanism of packaging in bacteriophage φ — 29 has recently been studied (Simpson et. al, Nature (London), 408, 745 (2000)). The study revealed the structure of the molecular motor that packages the DNA. In another experimental study, Smith et. Al
(Nature (London), 408, 745 (2001)) have investigated the effect of applied external force on the packaging. Motivated by this study, we suggest (K. P. Santo and K. L. Sebastian, Phys. Rev. E, 65, 052902 (2002)) a simple model to explain the kinetics of packaging of DNA the external force, which tries to prevent it. The model suggests a ButlerVolmer kind of dependence of the rate of packaging on the pulling force. We find that our model explains the experimental data very well.
Another very interesting situation that arises in biological contexts is the translocation of a polymer across a membrane through a pore. The uptake of DNA into the cell nucleus and the translocation of cytosolic protein into endoplasmic reticulum are examples. There have been two main classes of polymer translocation problems; translocation in presence of a field or driven by a molecular motor and the translocation assisted by the adsorption of molecules onto the chain in the region into which it is translocated. While the first class of problems is reasonably well understood, for the second class of problems a complete understanding still does not exist in the literature. The existing understanding of this kind of polymer translocation is mainly due to Simon, Peskin and Oster (Proc. Natl. Accad. Sci. USA, 89, 3770 (1992)), who describe the translocation as kind of biased Brownian motion, which is known as the Brownian Ratchet. But Brownian Ratchet is an idealization and can only be realized in certain limits and therefore, it does not account for the detailed dynamics of polymer and the binding particles. We present a simple statistical description of the problem. We find that in the regime where number of binding particles are larger than the number of adsorption sites on the chain, the translocation proceeds as if it is driven by a constant force and hence, seems to be governed by a mechanism similar to the kink mechanism (K. L. Sebastian and Alok. K. R. Paul, Phys. Rev. E, 62, 927 (2000), K. L. Sebastian, 61, 3245 (2000)) that has been suggested in the case translocation in presence of an external field. In the other regime, where the number of binding particles are less than the number of binding sites on the chain, the translocation was found to be predominantly diffusive.
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