Analytic Expressions

In this chapter, we will mostly be concerned with a system of a single helical fil- ament coiled around itself, as presented on Figure 3.1. This allows us to access lateral overlaps without a second filament, and it should yield contractility.

We start by considering the case of the single helical filament having length Lf with turn length L. The region where there is a lateral overlap

of that filament with itself is, then, Lf −L. We will indicate the number of

possible binding sites for cross-linkers as

l = Lf −L

δ (3.2)

where δ is the length of each binding site. In this way, we are able to render this quantity nondimensional.

We will assume that cross-linkers do not bind a side at a time, but rather at both sides at the same time. It makes the binding process one-step as opposed to two-step, and simplifies both the analytical expressions as the computational algorithm. This assumption has been shown to be valid for

Ase1 [Lansky et al., 2015], and we will assume it to be the case in our model system as well.

Furthermore, we will assume that this cross-linker binding process is non-cooperative; again, this is an assumption that is valid for Ase1, and we will consider it to be the case for our model. With that assumption, we are able to consider each binding or unbinding event as independent from all the others.

We are interested, initially, in writing down the expression for the over- lap expansion force in this system. In one dimension, we can write the one- dimensional equivalent of pressure associated with this system as

P = ∂E ∂L (3.3) which we approximate by P = ∆E ∆L = ∆E δ (3.4)

Therefore, for constant internal energy in the system, the force driving an increase in overlap length (which is, in this case, the contractile force) can be written as

F = ∆S

δ (3.5)

this depends only on the change in entropy associated with the change in overlap length. Finally, we can write the force expression as

F = kBT δ ln Z(l+ 1, n) Z(l, n) (3.6) where Z(l, n) is the partition function for n cross-linkers in an overlap with l

binding sites; for a system where n is constant, that is simply the number of ways then cross-linkers can be organized in the lbinding sites and be written as a combinatorial expression; that is,

Z(l, n) = l n = l! (l−n)!n! (3.7)

filament over the helical region of interest; increasing the curvature 2π/L de- creases the size of the region of interest and, therefore, translational freedom of the filament. This will be proportional to the circumference sizeL and we will see that the proportionality constant will not matter; therefore, we will take the partition function to be

Z(L, l, n) = L l n =L l! (l−n)!n! (3.8)

Thus, we can rewrite our force to depend only onl, the number of binding sites in the overlapping region before expansion, n, the number of cross-linkers in this region, and L, the circumference size that defines the filament curvature, by simplifying the factorial expressions in Z and arriving at

F(l, n) = kBT δ ln (L−1) L (l+ 1) (l+ 1−n) (3.9) where an increased overlap region comes at the expense of decreasing the circumference fromLtoL−1 and, since (L−1)(l+1)≥L(l+1−n) for almost all cases (the exception is a single cross-linker andl =L), the existence of even a single cross-linker in the overlapping region is already enough to generate a positive constrictive force.

We consider the case where L is large, so L ≈ L−1. As previously stated, that force depends only on the number of monomers in the overlap and the number of cross-linkers in the overlap:

F = kBT δ ln l+ 1 l+ 1−n (3.10) It is clear that the maximum force that can be generated is, therefore, the case therel =n, where we have

Fmax =

kBT

δ ln [n+ 1] (3.11)

By using the appropriate value ofkBT inpN.nm and the value ofδ= 5nmfor

the size of an FtsZ monomer, we could, therefore, calculate the hypothetical contractile force in piconewtons generated by a number of cross-linkers in a lateral overlap of a FtsZ filament, our model system.

Fmax =

4.114

5 ln [n+ 1]pN (3.12)

The maximum force of a system with constant number of cross-linkers, there- fore, scales with ln(n), leaving the scale of the force predominantly dominated by the prefactor.

The next point of interest is the force-velocity relationship in this sys- tem; Lansky et al. [Lansky et al., 2015] show that, for a constant number of cross-linkers, this is a linear relationship of the form

v = F

γM T

(3.13) whereγM T is an effective friction coefficient of the form

γM T =

nγ

2 (3.14)

where the factor of 2 accounts for the fact that γ is the drag coefficient be- tween a single head of the cross-linker and a filament. We assume cross-linkers to exclude each other, effectively implementing a sliding behaviour by taking cross-linkers to have an ordering associated with them. Finally, we can obtain the coefficientγ by Einstein’s relation kBT /D, where D is the diffusion coef-

ficient of a single cross-linker over a single filament. Thus, we can write our velocity expression as

v(l, n) = 2F

nγ (3.15)

and we can, then, replaceF by the previously calculated expression in Eq. (3.9) and γ by Einstein’s relation, yielding

v(l, n) = kBT δ D nkBT ln (L−1) L (l+ 1) (l+ 1−n) (3.16) and, finally, that can be simplified to

v(l, n) = 2D nδ ln (L−1) L (l+ 1) (l+ 1−n) (3.17) which depends on the diffusion coefficient of a single linker over a single fila- ment rather than on the more complicated factorγM T.

Now, we want to calibrate the results from our model, using the cal- culation of forces and velocities as specified above, versus a one-dimensional equivalent of the ideal gas law. To do that, we will compare the equilibrium point of the system from our simulations to what would be achieved through the ideal gas law.

Therefore, we need to be able to calculate what the equilibrium radius (equivalent to cell size) of a contracting helical filament would be, given its bending rigidity, initial size, intrinsic curvature and the number of cross-linkers in the system, based on the ideal gas law.

We will define the lengthL= 2πR, where R is the radius of curvature of the filament. We define the bending energy H following the Canham-Helfrich Hamiltonian [Helfrich, 1973] as H = 1 2 Z S κ 1 R −c0 2 ds (3.18)

whereκ is the bending rigidity of the filament, which means that

H = π 2Lfκ 2π L − 1 L0 2 (3.19) where we are approximating R as a constant around the whole filament for simplicity. Here, Lf is the length of the filament and L0 = 1/c0 is the inverse

of the preferred curvature of the filament.

From the one-dimensional lattice gas law, we have that the force gen- erated by the cross-linkers is

Fs =kBT

N Lf −L

(3.20) wheren is the number of cross-linkers andLf −L is the length of the overlap

region.

At equilibrium length, we have that the contractile entropic force from the cross-linkers should balance the force generated by the filament bending. Therefore, we have

Fs =−

∂H

This mechanical equilibrium condition is a third-order polynomial in L, that can be analytically solved for the equilibrium length by taking the real (or positive) root. We will present our findings in the Results section.