A statistical mechanics model for determining the length distribution of actin filaments under cellular tensional homeostasis

Tensional homeostasis is a cellular process whereby nonmuscle cells such as fibroblasts keep a constant level of intracellular tension and signaling activities. Cells are allowed thanks to tensional homeostasis to adapt to mechanical stress, but the detailed mechanism remains unclear. Here we address from a theoretical point of view what is required for maintaining cellular tensional homeostasis. A constrained optimization problem is formulated to analytically determine the probability function of the length of individual actin filaments (AFs) responsible for sustaining cellular tension. An objective function composed of two entropic quantities measuring the extent of formation and dispersion of AFs within cells is optimized under two constraint functions dictating a constant amount of actin molecules and tension that are arguably the two most salient features of tensional homeostasis. We then derive a specific probability function of AFs that is qualitatively consistent with previous experimental observations, in which short AF populations preferably appear. Regarding the underlying mechanism, our analyses suggest that the constraint for keeping the constant tension level makes long AF populations smaller in number because long AFs have a higher chance to be involved in bearing larger forces. The specific length distribution of AFs is thus required for achieving the constrained objectives, by which individual cells are endowed with the ability to stably maintain a homeostatic tension throughout the cell, thereby potentially allowing cells to locally detect deviation in the tension, keep resulting biological functions, and hence enable subsequent adaptation to mechanical stress. Although minimal essential factors are included given the actual complexity of cells, our approach would provide a theoretical basis for understanding complicated homeostatic and adaptive behavior of the cell.

As described in the main text, ! is the number of actin monomers in the cell involved in constructing AFs of length ! . Thus, ! / ! represents the number of individual AFs of length ! . The diameter of each molecule is smaller than a that is the compartment size, indicating that the number of compartments that can be occupied by a single actin monomer (or an AF of length % ) is always unity as the monomer is the smallest countable unit, which determines the first term in the middle expression of Eq. (8). Thus, i = 1 represents a group of AFs with the shortest length. As described in the main text, the diameter and filament lengths of actin molecules are normalized by the unit length % = 1. The compartment size of a = 1 means that the diameter of actin molecules is equal to the length of each compartment; thus, the number of compartments occupied by a single AF with a length of, e.g., ! = 10 is also 10.
Let us consider how the occupied compartments are counted in general with a > 1 based on some examples shown in Fig. 1b where a case with a = 3 % = 3 is drawn. Here, the maximum number of compartments that can be occupied by two ( & ), three ( 0 ), and four ( 1 ) bound actin molecules is all 2 as indicated by the gray-shaded area in the figure. If actin polymerizes to have a length of 2 composed of five actin monomers, the maximum number of compartments occupied by this AF is 3 as again indicated by the gray-shaded area. The count for these 2 and 3 consecutive compartments corresponds to the second and third terms in the middle expression of Eq. (8), respectively. Here, the number of the AFs is also considered; namely, in the case of the population of i = 5, the number of the AFs is 2 / 2 = 15/5 = 3. The general form, in which the maximum number of the compartments occupied by AFs is counted in the same way, corresponds to the right expression of Eq. (8), where m is, again, a group of AFs with the longest length. The partial derivative of Eq. (13) with respect to ! is The objective function ℒ takes an extreme value at which provides the existence probability for the present purpose Next, to determine the Lagrange multipliers α and β, Eq. (S5) is substituted partially into Eq.
(11) for the overall entropy, yielding The partial derivative of Eq. (S7) with respect to the expected value of force is With Eqs. (1) and (2), the summation of ! shown by Eq. (S5) must be unity: The partial derivative of Eq. (S9) with respect to is With Eqs. (1), (2), and (S5), Eq. (S10) is reduced to and therefore With Eq. (S8), the partial derivative of the overall entropy with respect to is suggesting that the Lagrange multiplier introduced to find the optimal distribution, α, can be expressed using the overall entropy and force expectation value.
Next, we discuss the specific form of α given that the entropy derived from a microscopic point of view based on statistical mechanics, maximizing the objective function at equilibrium, coincides with that derived from a macroscopic point of view based on thermodynamics. For the thermodynamic consideration, we consider a cell with a volume of V and a preexisting strain of λ. As we already discussed, the presence of the preexisting tension (or concomitant mechanical strain) is aimed at capturing the feature of tensional homeostasis. The strain varies spatially and temporarily at the microscopic view, while at the macroscopic view we consider an average level of strain distributed throughout the cytoplasm. The expected value of force introduced in Eq. (3) is regarded as stress in the following thermodynamic model.
The first law of thermodynamics gives where , , and denote the internal energy, the heat given to the system, and the work done to the system, respectively. Let us assume an elastic relationship between stress and strain λ, i.e.,

= E (S15)
where E denotes the elastic modulus of the cell, and it turns out that the strain energy is and accordingly the whole strain energy is described using volume V to be The work caused by change in stress, which in other words is the change in strain energy, is therefore in which cell volume is assumed to be constant (i.e., Poisson's ratio = 0.5).
The strain or intracellular deformation is generated within cells at tensional homeostasis upon the actin-myosin II interaction using the energy obtained from the ATP-mediated chemical reaction. Therefore, the strain energy discussed here is not the result of work done "by external forces" as in the case of conventional spring elasticity, but is the result of work done "by the system itself." In other words, as the cell system we discuss here includes subcellular parts that do work (namely, generate force and deform), the work done to the system must be opposite in sign to that done "by the system itself." It also turns out that, in the present system, the work done "by the system itself" will increase in amount as the strain/deformation-associated intracellular stress increases. Therefore, change in work of the entire system is With this, Eq. (S14) turns out where is the thermodynamic temperature. The free energy is defined as and its change is or with Eq. (S20) Describing the total differential form of ( , ), From Eqs. (S23) and (S24), The partial derivative of the free energy with respect to at constant temperature yields From Eqs. (S25) and (S26), At constant temperature where there is no change in internal energy, Eq. (S27) is reduced to describing the effect of intracellular stress on entropy derived from a macroscopic thermodynamic point of view. This positive relationship -distinct from the negative one for a conventional spring where stretch reduces the extent of fluctuations, the number of possible microstates of the constituents, and thereby the entropy as well -is reasonable because the stress of current interest is, as already mentioned, originated from the intracellular actinmyosin II interaction. More specifically, is interpreted to be an indicator of the activity of the intracellular components rather than a factor compelling their movement. Thus, an increased and resulting activation of the actin-myosin II interaction stabilize the cell structure as well as increase the entropy. Given that Eqs. (S13) and (S28) are equal at equilibrium, To determine , the constant part of Eq. (S30), which is independent of i, was expressed as B, and then With Eqs. (1) and (2), the summation of ! must be unity so that indicating that and β = −k $ N(lnB + 1).