In the previous sections we discussed various approaches for systems that are not in equilibrium.
Each approach was based on some assumption or the other. For example, in the case of Onsager relations we had to assume that the assumption of local equilibrium was right and that the system was close to equilibrium. When we use Langevin equation or the modified Langevin equation, we have to assume certain properties of the thermal bath. On the other hand, there exists extremely general results which are true no matter how far from equilibrium the system is. One such results is the non equilibrium work theorem, more popularly know as the Jarzynski’s equality. The content of the theorem is as follows.
Consider a system in equilibrium with the heat bath at temperatureT. Letλbe the extensive parameter of the system. For example, if the experiment involves pulling on a polymer, thenλwill be the distance between the two ends of the polymer. Let us say that the value of this parameter is λ1when the system is in equilibrium in state-1. Now, vary parameter λas a function of time from its initial value λ(0) =λ1 to its final valueλ(T) =λ2 with a prescribed time-history λ(t). Repeat the same procedure, say,N number of times. If the work obtained duringith trial isWi, Jarzynski’s equality [32, 33, 34] states that,
lim
N→∞
1 N
N
X
i=1
exp −Wi
kBT
=
exp
− Wi
kBT
= exp
−F2−F1
kBT
. (2.32)
Here F1 and F2 are the free energies of the system when the values of the extensive parameterλ are λ1 andλ2, respectively. This equality is true no matter what the time history of loadingλ(t) is. Apart from being a really nifty theorem in non-equilibrium statistical mechanics, the major attraction of Jarzynski’s equality lies in the fact that it can be utilized to obtain the free energy landscapes of biological systems [35]. Since free energy is an equilibrium quantity, in order to obtain the energy landscape for a system one needs to perform a time-consuming quasi static process.
The drift of the optical table, changes in temperature, vibrations, etc., make it difficult to perform
a real experiment on such time-scales. Similarly, the inherent limitations in molecular dynamics computation relative to the amount of time it can simulate makes it almost impossible to perform an equilibrium computational experiment. The Jarzynski’s equality, on the other hand, allows any rapid time history for the processλ(t). This makes the free energy estimation via real experiment, or by computation much more feasible.
The fluctuation theorems [36, 37] are a group of relations that describe the entropy production of a finite classical system coupled to a constant temperature heat bath, that is driven out of equilibrium by a time dependent work process. Although the type of system, range of applicability, and exact interpretation differ, these theorems have the same general form [36],
P(+σ)
P(−σ) 'exp(τ σ). (2.33)
Here,P(+σ) is the probability of observing an entropy production rate,σ, measured over a trajectory of time τ. This theorem thus quantifies the probability of violation of the second law of thermo- dynamics. Entropy production is an extensive quantity and depends on the size of the system, so larger the system, larger is σ, and the probability that entropy production is negative vanishes.
Similarly, the larger the time of observation τ, larger is the right hand side of Eq. 2.33, and hence the probability that entropy is consumed becomes negligible. The second law of thermodynamics is, hence, almost deterministically valid for large systems and large observation times.
Chapter 3
DNA Packaging and Ejection in Bacterial Viruses (Bacteriophage)
Most of the work in this chapter is published in Purohit et al. [28], and Grayson et al. [13]
The conjunction of insights from structural biology, solution biochemistry, genetics, and single molecule biophysics has provided a renewed impetus for the construction of quantitative models of biological processes. One area that has been a beneficiary of these experimental techniques is the study of viruses. In this chapter we describe how the insights obtained from such experiments can be utilized to construct physical models of processes in the viral life cycle. We focus on dsDNA bacteriophages and show that the bending elasticity of DNA and its electrostatics in solution can be combined to determine the forces experienced during packaging and ejection of the viral genome.
Furthermore, we quantitatively analyze the effect of fluid viscosity and capsid expansion on the forces experienced during packaging. Finally, we present a model for DNA ejection from bacterio- phages based on the hypothesis that the energy stored in the tightly packed genome within the capsid leads to its forceful ejection. The predictions of our model can be tested through experiments in vitro where DNA ejection is inhibited by the application of external osmotic pressure.
As shown in Fig. 3.1, a typical phage life cycle consists of adsorption, ejection, genome replication, protein synthesis, self-assembly of capsid proteins, genome packaging inside the capsid, and lysis of the bacterial cell. A wealth of knowledge about various aspects of these processes has been garnered over the last century, with the main focus being on replication and protein-synthesis. However, recent developments in the fields of x-ray crystallography, cryo-electron microscopy, spectroscopy, etc., have helped reveal the structural properties of the viral components [38, 39] involved in ejection, assembly, and packaging, while recent experiments on φ29 [11] and λ [12] have helped quantify the forces involved in the packaging and ejection processes, respectively. In this chapter we bring together these experimental insights to formulate quantitative models for the packaging and ejection processes. The model is based on what we know about the structural “parts list” of a phage: the shape, size, and strength of the capsid the mechanical and electrostatic properties of DNA and
high-resolution images that reveal the structure of assembled phage particles. Our goal is to create a detailed picture of the forces implicated in DNA packaging and ejection and, most importantly, make quantitative predictions that can be tested experimentally.
Figure 3.1: Life cycle of a bacterial virus. The ejection of the genome into the host cell happens within a minute for phage like λ and T4 [14, 40]. The eclipse period (time between the viral adsorption and the first appearance of the progeny) is about 10-15 minutes [41]. The packaging of the genome into a single capsid takes about 5 minutes [11]. Lysis of the bacterial cell is completed in less than an hour [41].
The chapter is organized as follows: In Section 5.1 we provide a brief introduction to viruses in general, and bacteriophages in particular. In Section 3.2 we examine dsDNA bacteriophage with the aim of assembling the relevant insights needed to formulate quantitative models of packing and ejection. In Section 3.3 we develop the model by examining the DNA packaging process in detail, and in Section 3.4 we show that it can be used to explain the DNA ejection process as well. In Section 3.5 we take stock of a range of quantitative predictions that can be made on the basis of the model and suggests new experiments.