Appendix

RRout

4.6 Appendix

4.6.1 Justification of the Expression for Steady-state First Passage Time

To measure the mean first passage time (MFPT) for a particle starting at an impenetrable wall at x = 0 and reach x = L₀, consider the following though experiment. Take an experimental system containing a large numberM of non-interacting particles in a box of size L₀. Let the setup of the experiment be such that no sooner than a particle reaches L₀, it will be scooped and put back at x = 0. This is to ensure that the total number of particles in the box is constant, i.e., RL0

0 p(x, t)dx= 1. Here p(x, t) =c(x, t)/M is the probability of finding a particle at position xat timetandc(x, t) is the concentration of particles in the box. Iftiis the time after which a particle is seen to arrive at L0 during theith measurement, the mean-first passage time, by definition, will be given by

t₀= lim

N→∞

PN i=1t_i

N , (4.11)

where N in the denominator represents the total number of particles encountered during the time PN

i=1ti. Since the system is “non-driven,” and the total number of particles in the system is constant, we will be into the steady state regime after some finite timetk. Now, if we continue our measurement well beyond that time we can split the above equation as follows:

t₀= lim

N→∞

Pk

i=1ti+PN j=k+1tj

k+ (N−k) (4.12)

Now, if we denotePk

i=1t_k ast, andPN

i=k+1t_i asT, the equation for MFPT becomes t0 = lim

N→∞

t+aT

k+ (N−k) (4.13)

= lim

N→∞

T (N−k)

1 +t⁰/T 1 +k/(N−k)

(4.14)

= lim

N−k→∞

T

(N−k). (4.15)

But the last equation is simply 1/j0, wherej0is the constant steady-state flux. Thus the MFPT for a particle to start atx= 0, and reachx=L0 is equal to the reciprocal of the steady state fluxj0.

That relation between the MFPT and the steady-state flux was a completely general result. We now wish to evaluate MFPT for the diffusion of a particle on an energy landscape U(x). If we assume that this particle diffusion obeys a Fokker-Planck equation, the steady-state flux will be given by [112]

j₀=−D

∂p(x)

∂x + 1 k_BT

∂U(x)

∂x

. (4.16)

This can be rewritten as d dx

p(x) exp(U(x) kBT )

=−j0

Dexp U(x)

kBT

. (4.17)

Now, if we integrate this equation from the limitsxtoL0, we will obtain p(L₀) exp

U(L₀) kBT

−p(x) exp U(x)

kBT

=−p(x) exp U(x)

kBT

=−j₀ D

Z x0

x

exp U(x)

kBT

. (4.18)

We have used the fact that because we are interested in the MFPT, as described above, we absorb the particle soon as it reachesx=L0, as a result of whichp(L0) = 0. Multiplying both sides of the above equation by−exp (−U(x)/kBT) and integrating from 0 toL0 we get

j0

D Z L₀

0

dxexp

−U(x) k_BT

Z L₀ x

exp U(y)

k_BT

dy= Z L₀

0

p(x)dx= 1. (4.19) Upon rearrangement we get

t0= 1 j0

= 1 D

Z L₀ 0

dxexp

−U(x) kBT

Z L₀ x

exp U(y)

kBT

dy. (4.20)

This is the precisely the equation (Eq. 4.4) used in Section 4.3 of the chapter.

4.6.2 Donnan Equilibrium Considerations When the Virus Injects DNA into the Vesicle

The difference between the vesicle and the viral capsid is that the viral capsid is permeable to the ions in the surrounding solution, while the vesicle is not. On the other hand, when we model the DNA inside the vesicle we use the same expression for the internal energy obtained from the experiments of Rau et al. [47], which essentially have free mobility of ions between the DNA condensate and the surrounding solution. How can we justify doing this? To do so we will first pretend that the vesicle is permeable to the surrounding ions. We then assume that the concentration of the ions inside the vesicle is as a result of Donnan equilibrium [120, 75] between the ions/DNA inside the vesicle and the ions in the surrounding solution. We then use a reverse process to find out what the concentration of the surroundings solution should be so that it is in equilibrium with the ions/DNA inside the vesicle, whose concentration is known.

Suppose the solute (corresponding to the experiments by Novick and Baldschwieler [14]) in the vesicle is MgCl2. We first balance the chemical potentials of these ions inside the vesicle with with those in the surrounding solution. The chemical potential for the Cl⁻¹ and Mg⁺² ions inside the vesicle is given by

µ_Cl⁻¹

I = µ⁰_Cl−1 I

−eVI[Cl⁻¹_I ] + RT ln [Cl⁻¹_I ] [Cl⁻¹]0

and (4.21)

µ_Mg+2

I = µ⁰_Mg+2 I

+ 2eVI[Mg⁺²_I ] + RT ln [Mg⁺²_I ]

[Mg⁺²]₀, (4.22)

where eis the charge of an electron,VI is the effective potential inside the vesicle due to the ions, and [Mg⁺²]0and [Cl⁻¹]0are the so called standard concentrations. Similar equations can be written for the ions outside the vesicle.

µ_Cl−1 O

= µ⁰_Cl−1 O

−eV_O[Cl⁻¹_O ] + RT ln [Cl⁻¹_O ] [Cl⁻¹]0

, and (4.23)

µ_Mg+2

O = µ⁰_Mg+2 O

+ 2eVO[Mg⁺²_O ] + RT ln [Mg⁺²_O ] [Mg⁺²]0

, (4.24)

Since the chemical potentials of similar species are equal at equilibrium, equating Eq. 4.22 with Eq. 4.24 and Eq. 4.22 with Eq. 4.24, we obtain,

RTln[Cl⁻¹_O ]−eVO = RT[Cl⁻¹_I ]−eVIand (4.25) RTln[Mg⁺²_O ] + 2eVO = RT[Mg⁺²_I ] + 2eVI. (4.26)

Multiplying Eq. 4.25 and adding it to Eq. 4.26 we obtain,

[Mg⁺²_O ][Cl⁻¹₀ ]²= [Mg⁺²_I ][Cl⁻¹_I ]². (4.27) We also need to have charge neutrality both inside and outside the vesicle. This will give us

2[Mg⁺²_O ] = [Cl⁻¹_O ], (4.28)

z[DNA] + [Cl⁻¹_I ] = 2[Mg⁺²_I ]. (4.29) We can combine these expressions for the chemical potentials with that of the charge neutrality and get

4[Mg⁺²_O ]³= [Mg⁺²_I ](2[Mg⁺²_I ]−z[DNA])², (4.30) where z is the negative charge on the DNA, which is 2 per bp. It can be seen from this equation that when the DNA concentration [DNA] is zero, there is no difference between the external and the internal concentration of the magnesium ions. When the DNA concentration increases the difference between the external and the internal concentration of the ions will increase. As a conservative estimate, we wish to find the highest possible concentration of the DNA inside the vesicle so that we get the maximum possible change in the external concentration. For that we take minimum vesicle radius 30 nm used in Section 4.3. The volume of this capsid will be 10⁻¹⁹ liters. The total charge on the 48.5 kbp λDNA is 2×48500 ∼10⁵. Hence, the charge concentration due to the DNA in

“molar units” M is 10⁵/(10⁻¹⁹NA)∼2M. Before the DNA entered into the vesicle the vesicle was charge neutral. To maintain that charge neutrality the DNA brings its own Mg⁺² counterions with it into the vesicle. So if the concentration of M g inside the vesicle before the entry of DNA is 10 mM, the concentration after the entry will bez[DNA]/2 + 10⁻² M. Substituting this into Eq. 4.30 we obtain

4[Mg⁺²_O ]³ = (z[DNA]/2 + 10⁻²)(2(z×[DNA]/2 + 10⁻²)−z[DNA])²M³ (4.31)

=⇒[Mg⁺²_O ] = (10⁻⁴(z[DNA]/2 + 10⁻²))^1/3M. (4.32) We already evaluate the maximum value ofz[DNA] and from this we find the concentration of Mg⁺²_O is approximately 50 mM. But we know from the experiments of Rau et al. [47] that within the range of 1mM-100mM for Mg⁺²_O the force measurements give almost the same results. So it would be safe to assume that vesicle is like the viral capsid and is permeable to ions (with the same salt strength in the surrounding).

Chapter 5

DNA Fluctuations in Nucleosomes

We saw in Chapter 1 that the eukaryotic genome is much longer than the characteristic size of a cell and needs to be compacted so as to fit inside the nucleus. In order to effect this compaction, the eukaryotic genome is organized into complex structures called nucleosomes, which consist of DNA tightly wrapped around four pairs of proteins called histones. In Chapter 1 we also saw that in order to perform important functions like gene expression, DNA repair, DNA replication, etc., different parts of the genome must be physically accessible to various proteins at different times.

Since the eukaryotic DNA is effectively sequestered by virtue of packing within nucleosomes, at first thought it would seem that the genome is inaccessible, but it must be noted that the nucleosome is not a static structure, and it is highly possible that the genome accessibility is wrought out by the thermal fluctuations of the nucleosomal DNA involving its unwrapping and re-wrapping around the histone core. Although the in vivo case may be much more complicated, recent ingenious experiments [24, 22, 23] under in vitroconditions have demonstrated the role of these fluctuations and have measured the equilibrium constant and the rates for the DNA wrapping and the re- wrapping processes (see Fig. 1.3). In this chapter we model the DNA-histone system by taking into account the DNA bending elasticity and the DNA-histone interaction. We make predictions for the equilibrium constant associated with the wrapping and re-wrapping processes using the standard grand canonical ensemble of equilibrium statistical mechanics [26]. We also provide estimates of the DNA wrapping and re-wrapping rates using a Fokker-Planck description. In summary, we make quantitative predictions of these DNA fluctuations as a function of the length of the DNA under consideration with special reference to the probability of unwrapping the nucleosome by a given amount.