The Equilibrium Constant and the Unwrapping and Re- wrapping Rates as a Function of the Depth of the Buriedwrapping Rates as a Function of the Depth of the Buried

RRout

5.4 The Equilibrium Constant and the Unwrapping and Re- wrapping Rates as a Function of the Depth of the Buriedwrapping Rates as a Function of the Depth of the Buried

Sites

In Section 5.2 we discussed briefly the experiment by Polach and Widom [24]. In this section we will formulate a model using statistical mechanics to provide us with the value of the configurational

equilibrium constantK_eq^configas a function ofz, the distance of the binding site from the edge of the nucleosomal DNA. To do so, consider a nucleosome with a specific binding site at a distancez from the nucleosome edge and surrounded by a reservoir of the corresponding binding proteins at chemical potentialµ. We assume that the nucleosome/binding protein system has reached equilibrium. The binding protein can bind on the DNA only when the site at distancez is exposed. As discussed in Section 5.1, it was shown by Li and Widom [22] that the DNA sequentially unwraps from one end, as opposed to forming bubbles. As a result, the enumeration of the equilibrium states and, hence, the evaluation of the partition function becomes quite simple. The total number of microstates fall under two categories: (i) when the binding protein is not bound to the DNA; (ii) when the binding protein is bound to the DNA. Now, if we assume that the DNA unwraps only from one side, the amount of DNA exposed xcan take values between 0 and L. Hence, the total number of possible microstates when the binding protein is not bound to the DNA is given by the values ofxbetween 0 and L. The binding protein can bind on to the DNA only when at least z length of DNA is unwrapped. The total possible microstates when the binding protein is bound to the DNA is x betweenz and L. The partition function is the summation of the Boltzmann weights for all these microstates. The partition function [26] then can be written as

Z= Z L

0

exp(−¯α(L−x))dx+ exp

µ+bind

k_BT

Z L z

exp(−¯α(L−x))dx, (5.19)

where ¯α = α/kBT, α = γ+ξpkBT /2R² is as discussed in the previous section and bind is the lowering of the energy when the protein binds to the binding site. The first integral in the partition function corresponds to the case when the protein isn’t bound on the DNA, while the second integral corresponds to when at least length z of the DNA has unwrapped and the protein is bound on to the DNA. The extra exponential in the second integral involving sum of the chemical potential µ and the binding energybind is the effective binding free energy when the protein binds on to the DNA atx=z. The partition function can be easily evaluated as

Z = 1

¯ α

1−exp(−¯αL) + exp

µ+_bind kBT

[1−exp(−¯α(L−z))]

. (5.20)

The fraction of nucleosomes bound with the binding proteins is given by θ = ∂lnZ

∂µ and (5.21)

=

(R) (R₀)exp

_bind k_BT

[1−exp(−¯α(L−z))]

1−exp(−αL) +¯ _(R^(R)

0)exp

_bind kBT

[1−exp( ¯α(L−z))]. (5.22)

We have used the standard ideal-solution result [122] that exp(µ/k_BT) = (R)/(R₀), where, as in Section 5.2, (R) is the concentration of regulatory proteins in solution, while (R₀) is referred to as the standard concentration. As described in Section 5.2, the fraction of nucleosomes bound can also be expressed in terms of the equilibrium constantK_d^apparent, which gives the ratio of the product of concentrations of free proteins in solution (R) and unoccupied binding sites (S), to the concentration of binding sites occupied by regulatory proteins (RS):

K_d^apparent= [(N) + (S)](R)

(RS) . (5.23)

The fraction of nucleosome bound by the proteins can be expressed in terms of concentrations of occupied and unoccupied sites as

θ= (RS)

(RS) + [(N) + (S)]. (5.24)

Using Eq. 5.23, we get

θ= (R)/K_d^apparent

1 + (R)/K_d^apparent. (5.25)

Comparing Eq. 5.22 with Eq. 5.25 we obtain K_d^apparent= (R0)

exp (_bind)

1−exp(−¯αL)

1−exp(−¯α(L−z)). (5.26)

We can split this equilibrium constant intoK_nakedandK_eq^config, referring to the equilibrium constant just in the presence of the binding sites without the wrapping and unwrapping, and the equilibrium constant associated with simply the wrapping and unwrapping, respectively. As seen in Section 5.2 we can think of binding as occurring in two steps. The first step corresponds to a transformation of a binding site which is hidden by the presence of the histone to one that is exposed due to DNA unwinding. Since the concentration of exposed sites is (S), the equilibrium constant for this process is

K_eq^config = (S) (N) and

=⇒ K_eq^config 1 +Keq^config

= (S)

(N) + (S). (5.27)

This equilibrium constant K_eq^config is independent of the strength of the binding site and depends just on the histone-DNA energetics. The second step of the binding process is the binding of the

regulatory protein to an exposed binding site. The equilibrium dissociation constant in this case is

K_d^{naked DNA}= (R)(S)

(RS) . (5.28)

Dividing Eq. 5.28 with Eq. 5.27 and comparing it with Eq. 5.23 we obtain

K_d^apparent=K_eq^{naked DNA}K_eq^config+ 1 Keq^config

. (5.29)

It may be noted that if K_eq^config 1, the above equation reduces to Eq. 5.4 used by Polach and Widom [24]. It may also be noted that K_eq^{naked DNA} = R0/exp(bind/kBT. Hence, comparing Eq. 5.29 with Eq. 5.26 we obtain

K_eq^config 1 +Keq^config

= 1−exp(−¯α(L−z))

1−exp(−¯αL) and (5.30)

=⇒ K_eq^config = exp( ¯αz)×(exp( ¯α(L−z)−1)

(exp( ¯αz)−1) . (5.31)

The above equation is easy to interpret. The nucleosomal DNA is in the wrapped state,W, when the binding site at distancezis not exposed, i.e., the nucleosome is in stateW forx < z. The probability that this happens is the summation of the Boltzmann probabilities satisfying this condition,

p_W = 1 Z

Z z 0

exp(−¯α(L−x))dx and

= exp(−¯αL)

¯

αZ (exp( ¯αz)−1). (5.32)

On the other hand, when the binding site gets exposed, the nucleosome is in unwrapped state, U. This happens whenx≥z, i.e,x∈[z, L]. Hence, similar to the evaluation ofp_W as done in Eq. 5.32, the probability that the nucleosome is unwrapped is

pU = 1 Z

Z L z

exp(−¯α(L−x))dx and

= exp(−¯αL)

¯

αZ exp( ¯αz)×(exp( ¯α(L−z)−1). (5.33) The configuration equilibrium constant is, hence, obtained by dividing Eq. 5.33 with Eq. 5.32:

K_eq^conf= pU

p_W = exp( ¯αz)×(exp( ¯α(L−z)−1)

exp( ¯αz)−1 , (5.34)

which is the same as that obtained in Eq. 5.27. We thus have obtained the equation for K_eq^config(z) in terms of the microscopic parameters that involve the bending of the DNA and the DNA-histone interaction energy. In next section we will evaluate the numerical values ofK_eq^config(z) and compare

them with the experimental data of Polach and Widom [24]. Note that we have assumed that the unwrapping happens only from one end of the nucleosome. It will be shown in Appendix 5.7.1 that taking into account the unraveling of the DNA from both ends gives us almost the same result.

We now turn our attention towards obtaining the expressions for the rate constantk₁₂andk₂₁, as shown in Fig. 5.2 and Eq. 5.1. We first obtain an expression fork12, the unwrapping process. To do so we will make a few assumptions:

1. For convenience we will assume that the DNA unwraps only from one end of the nucleosome.

This may seem as a bit of a harsh assumption, but it is in accordance with the experimental conditions of Li et al. [23]. As explained in Section 5.2, Li et al. [23] used FRET to obtain the measurement ofk₁₂ by putting an acceptor dye Cy3 on one end of the DNA and two acceptor dyes Cy5 on the pair of histone protein H3. The FRET showed a signal loss only when the Cy3 end got separated from Cy5. Since there was no dye on the other end of the nucleosome, any amount of fluctuations of the DNA from the other end of the nucleosome would be immaterial in reducing the FRET signal. In short, the experimental conditions measured the fluctuations only from one end of the DNA. Of course, this is an experimental caveat in the sense that one can add a Cy5 dye also on the other end to characterize the fluctuations of that end. A more physical reasoning behind neglecting the fluctuations of the other end of the DNA is that if the binding site is closer to the first end of the DNA, it is much more unlikely for the binding protein to access that site by a fluctuation from the other end as compared to from the first end.

2. If xand L−xare the lengths, respectively, of the unwrapped and the wrapped part of the DNA, we assume that the DNA is in local equilibrium for everyx, i.e., the rate of unwrapping is slow compared to the local equilibration of the DNA-histone system. This assumption is to ensure that we can use the expression for free energy F(x), as shown in Eq. 5.18.

3. The dynamics of DNA unwrapping obey a Fokker-Planck [111](also see Chapter 2) description.

This means that we can model the unwrapping as the diffusion of DNA happening in the background of the free energyF(x) in Eq. 5.18.

4. The rate constantk12is equal to 1/τ12(x), whereτ12(x) is the mean-first-passage-time (MFPT) for the DNA to unwrapxnm of DNA. We have seen in Chapter 4 that, given a Fokker-Planck equation, the MFPT is easily obtained. We want to utilize this property of the Fokker-Planck equation. Also, MFPT seems to be the right description for the experimental measurement of k12, where the unwrapping was defined as the first time that the FRET loses significant amount of its signal (see Section 5.2 for more details).

Let us now model the kinetics of the unwrapping process. As mentioned earlier, we assume that

the DNA is in local equilibrium for every segment of lengthxunwrapped, i.e., the free energyF(x) is well defined. In that case, it is reasonable to model the unwrapping of the DNA as a diffusion-in- a-field problem. The diffusion variable here is the lengthxof the DNA unwrapped, and the external field is the free energyF(x). Similar to the DNA ejection process in bacteriophage (see Chapter 4), the problem reduces to the Fokker-Planck equation. It involves p(x, t), i.e., the probability thatx length of DNA will unwrap in timet, and is given by

∂p(x, t)

∂t =D

∂²p(x, t)

∂x² + 1 kT

∂

∂x

∂F(x)

∂x p(x, t)

. (5.35)

In this equation the quantityD is the diffusion constant for the unwrapping DNA and signifies the

“local” diffusion of the DNA. The MFPT (as in Chapter 4) is [112]

τ12(x) = 1 D

Z x 0

dx1exp

F(x1) kT

Z x₁ 0

dx2exp

−F(x2) kT

. (5.36)

The integral is very easy to evaluate because in our model the free energy F(x) (see Eq. 5.18) is linear inx. The expression for the MFPT as a function ofxmay be written as

τ₁₂(z) = 1

k₁₂(z) = 1 D

Z z 0

dx₁expαx1

kT Z x1

0

dx₂exp

−αx2

kT

and (5.37)

= 1

D kT

α ²

×(expαz kT

−1)−kT α z

!

, (5.38)

whereα=γ+ξkT /2R²is the total free energy per unit length of the wrapped DNA. Thus we have obtained an expression fork12(x). In the experiment by Liet al. the value of re-wrapping ratek21is found by using the relationshipk21=k12/Keqand knowingK_eq^configfrom [22]. We can use our model to make an estimate ofk12 without referring to the equilibrium constant. To do so, we only need to replaceαin Eq. 5.38 with−αbecause when the DNA re-wraps it undergoes the exact opposite process as of unwrapping. The expression fork21 is

τ21(z) = 1 k21(z) = 1

D kT

α 2

×(exp

−αz kT

−1) +kT α z

!

. (5.39)

The above estimate of re-wrapping relies on the additional assumption that, when the DNA re-wraps it does not do it haphazardly but retraces its original path.

We have seen in Section 5.1 that thermal fluctuations of the nucleosomal DNA is proposed as one of the methods, which explains how the binding sites on the genome are exposed to various proteins in order to perform many key genomic functions. The time window available for these proteins depends on the unwrapping and re-wrapping rates. Our theoretical expressions make falsifiable predictions about these time-scales and thus tell how feasible various genomic processes are depending on how

5 10 15 20 z nm

-6 -5 -4 -3 -2 -1

LogKeq

K

^conf_eq

k

₂₁

k

₁₂

a)

5 10 15 20

z nm -8

-6 -4 -2

LogKeq

γ = −0.65/nm

γ = −0.8/nm

Figure 5.4: The configurational equilibrium constant predicted by the theoretical model. a)The configurational equilibrium constant K_eq^const as a function of depth of the binding sitez. Best fit is obtained to the experimental data (dots in the plot) when γ = −0.65kBT. b)The comparison of the Equilibrium constant obtained using Eq. 5.31 and the one obtained using Eqs. 5.38 and 5.39, k12/k21. There is a clear discrepancy between the two results. The discrepancy may be attributed to the fact that unlike the bistable energy landscape of the chemical reactions obeying, which obey the principle of detailed balance [123], the linear landscapeF(x) (see Eq. 5.18) does not admit clear cut definition of “states.”

far into the nucleosome the binding sites are buried.