To illustrate the general theory discussed above, we consider a simple coupled harmonic spring system shown in Fig. 3.1. Two balls are attached to three springs with the same spring constant k. The system has two degrees of free- dom: x1 andx2being the displacements of the two balls from their equilibrium
positions. The Hamiltonian of the system is:
H= 1 2kx 2 1+ 1 2kx 2 2+ 1 2k(x1−x2) 2. (3.21)
The stiffness matrix constructed from Eq. 3.21 is:
M= 2k −k −k 2k (3.22) Hence, using Eq. 3.6, the fluctuation of the system is:
hxixji=kBT M−1 ij = kBT 3k 2 1 1 2 (3.23)
Figure 3.1: A simple coupled harmonic spring system with two balls attached to three springs. All three springs have the same spring constant k and natural lengthL. The degrees of freedom for this discrete system are two, characterized by x1 and x2, the displacements of the two balls away from their equilibrium
positions.
Now suppose we have an additional constraintx1 =x2, i.e, the two balls are
further connected by a rigid bar. The first method is to replace the constraint by a stiff spring kc and the effective Hamiltonian of the system becomes:
H = 1 2kx 2 1 + 1 2kx 2 2+ 1 2(k+kc)(x1−x2) 2. (3.24)
The stiffness matrix constructed from Eq. 3.24 is:
M= 2k+kc −k−kc −k−kc 2k+kc (3.25) So the fluctuation of the system is:
hxixji=kBT lim kc→+∞ M −1 ij = kBT 2k 1 1 1 1 . (3.26) This makes sense because the system with constraint x1 = x2 is just like a
rigid bar connected by two springs. The effective spring constant is 2k. For the method that uses Fourier transform of the δ-function, the effective Hamiltonian of the system is:
H = 1 2kx 2 1+ 1 2kx 2 2+ 1 2k(x1−x2) 2−Ik ckBT(x1−x2). (3.27)
The corresponding stiffness matrix constructed from Eq. 3.27 has three degrees of freedom: M = 2k −k −kBT I −k 2k kBT I −kBT I kBT I 0 (3.28)
The inverse of M multiplied by the thermal energy gives the fluctuation
hxixji = kBT /2k, which is exactly the same as the result using the previ-
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Chapter 4
Heterogeneous Wormlike Chain
Under End-to-end Force
Main results of this chapter:
1
Statistical mechanical properties of a general heterogeneous chain under various boundary conditions are evaluated using the multidimensional Gaussian integral technique.2
All thermodynamic properties of a chain can be expressed in terms of an energy function which is measurable in a single force-extension experiment. Formulae in terms of this energy function hold for any heterogeneous chain under any boundary condition.3
Folded and unfolded proteins are expected to have different mechanical properties. Here the theory for a heterogeneous chain with two bending moduli is applied to predict the force-extension profile for the forced unfolding of a protein oligomer.4.1
Introduction
Single molecule mechanical experiments on rod-like biomolecules, such as, DNA and actin have for long been interpreted using a model of a homoge- neous fluctuating elastic rod [1, 2, 3, 4]. However, advanced single molecule techniques are now capable of probing the structure and properties of macro- molecules at length scales of a few nanometers. At these length scales it is no longer sufficient to think of the molecules as having homogeneous mechanical
properties. In fact, several recent studies have revealed the remarkable effects of the heterogeneous properties of some biopolymers on their conformations as well as their mechanical behavior [5, 6]. For example, heterogeneous me- chanical properties are encountered in partially unfolded protein oligomers in atomic force microscopy [7]. Sequence specific mechanical properties of DNA are already well known and it has been suggested that DNA binding proteins can sense these heterogeneities, making them biologically significant [8, 9, 10]. Also, it has been noted that localized softening in DNA can have significant influence on looping probabilities which ultimately affect genetic activity [11, 12]. These examples show that heterogeneous mechanical prop- erties have been observed in experiments on biomolecules and that even at relatively large length scale, they can have significant biological consequences which the homogeneous models cannot caputre. They motivate us to examine the consequences of heterogeneity through detailed mathematical models.
A simple way of introducing heterogeneity in polymer models is to group monomers into hydrophilic and hydrophobic types as has been done in some recent articles [13, 14]. Another model which accounts for heterogeneity is the two-state wormlike chain model of [15], which reduces to the fluctuating rod model in the low force limit, and to the Ising model at high forces [15]. The approach in this chapter is different from these methods in that we allow the bending modulusKb(s) of our fluctuating rod to vary as an arbitrary function
of the arc length s. We first evaluate the partition function of the rod in a constant force and constant temperature ensemble, and then compute the free energy and a host of other thermal and mechanical properties of the rod. The results are verified through Monte Carlo simulations. A special case of our model is one in which there are only two possible values of the bending modulus KI and KII along the chain. We call this the ‘special heterogeneous
chain’ and use it to interpret the force-extension data from the forced unfolding experiments on protein oligomers.
Our method also allows us to determine the consequences of constraints imposed on the rod. In particular, we can determine the force-extension re- lation and the magnitude of transverse fluctuations under different types of boundary conditions. Boundary conditions siginificantly affect the fluctuations if the length of the rod is comparable or shorter than its persistence length [16]. The effect of boundary conditions on the fluctuations of homogeneous rods has been analyzed only recently by a few authors [17]. In this chapter we apply three different boundary conditions on the rod and compare our results with those of [17] for homogeneous rods and find excellent agreement. The method used in this chapter is more general than that of [17] which is based on the equipartition theorem and can only be applied to homogeneous rods.
This chapter is organized as follows. We first use the equipartition theorem to derive some general results for heterogeneous chains with arbitrary bound- ary conditions. We then demonstrate a method for calculating the thermo- mechanical properties of chains and rods under three different boundary con- ditions. We use Monte Carlo simulations and comparisons with earlier work
to show that our method gives accurate results. Finally, we apply our method to interpret data from force-extension experiments on the protein ubiquitin.