Mechanochemical regulation of oscillatory follicle cell dynamics in the developing Drosophila egg chamber

In the epithelium of Drosophila during tissue elongation, contractile forces in follicle cells can oscillate. These oscillations correlate with increasing tension in the epithelium from egg chamber growth. A mathematical model is proposed to explain the observed oscillations, together with a mechanism of active regulation of cellular contractile forces.


Equations Mechanical cell model
For a cross section of unit cell width in Fig. 1 in the main text, motions of the cell ends in the r and θ directions can be obtained from a mechanical energy formulation of the cell layer. This energy is a sum of the elastic energies -from follicle cells as well as the connectors to the basement membrane, the work done by the actomyosin contractile force and the work done by pressure P inside the egg chamber. The mechanical energy per length is then where r is the radius of the circular cell array and is assumed to be the same for all the cells, l o is the rest length of the cell, N is the number of cells in a cross section, N k is the effective stiffness of the basement membrane, r o is the preferred radius of the egg chamber. At the scale of the egg chamber, inertia is unimportant and forces are balanced by viscous friction. The equation of motion for r and θ i can be obtained from the mechanical energy by differentiation with respect to these variables and equating them to friction. The equations are then where η and γ represent the frictional coefficients in the θ and r directions respectively. Note that in the absence of cellular contractile forces, follicles cells are stretched by internal pressure, P . We propose that the cells generate contractile force that opposes the egg chamber pressure, and it is the biochemical control of the contractile stress that generate oscillations.

Biochemical model
Models of Rho-ROCK signaling pathway has been studied before [Jilkine et al., 2007]. Here we propose a model where the activation of Rho and myosin are related to mechanical tension in the cell. Increased activation of Rho, ROCK and MLC in tissue cells in response to external tension has been observed [Zhao et al., 2007, Bhadriraju et al., 2007. We propose that this is also present in follicle cells. The kinetics of Rho, ROCK and MLC activation in the i-th cell are modeled as where ρ i , R i , m i represent the fraction of activated Rho, ROCK and MLC respectively. s i is the change in length of the i-th cell, (r(θ i+1 − θ i ) − l o ) (the mechanical tension is then k c s i ). Every rate equation has an activation and a deactivation part. The deactivation part is linear, which represents self degradation. In the activation part, f ρ (s) represents effect of the mechanical tension on the activation of Rho. Similarly f R (ρ) and f m (R) represent effect of Rho on ROCK and the effect of ROCK on MLC respectively. Mathematically, to obtain sustained oscillations, non linearity in the system is essential. We incorporate non linearity in the form of a Hill function for the effect of tension on the activation of Rho, representing possible cooperativity in Rho activation: where h is a heaviside step function which is zero when s is negative and 1 when s is positive. This ensures that Rho gets activated upon cell stretching under tension. A ρ , A R, A m are the rates of activation and D ρ , D R, D m are the rates of deactivation. Activation of Rho by other stimuli (growth factors and cytokines) and activation of ROCK and MLC by other pathways [Burridge and Wennerberg, 2004, Buchsbaum, 2007, Burridge and Doughman, 2006] are assumed to be included in these rate constants. K s is the half maximal response constant and n is the Hill coefficient for cooperativity.
Since the contractile force F originates from the activation of myosin light chain (MLC), we can assume that the force scales linearly with the fraction of activated MLC which should be proportional to the accumulated myosin quantified in experiments. The proportionality constant F max represents the contractile force of a cell when the activated myosin fraction is 1, i.e, the maximum contractile force.
Note that the proposed mechanical signaling model explains why stress fibers and contractile force are in the D-V direction in follicle cells. For approximately cylindrical egg chamber, the mechanical tension from the internal pressure P is P R in the D-V direction and P R/2 in the A-P direction. Therefore, stress-fiber activation would occur in the D-V direction first. The internal pressure and the shape of the egg chamber determine the direction of oscillation.

Nondimensionalization
The equations above can be non dimensionalized by considering the characteristic length scale to be r o and the time scale to be γ kc . The time scale ( γ kc ) is in the order of a couple of minutes (100 seconds). The activation rates we use in the model are also in the same range. where Numerical values of the dimensionless parameters are as follows. D 1 ranges from 0.33 to 2, D 2 ranges from 83 to 89, D 3 is equal to 91.25, D 4 is equal to 1, D 5 to D 10 (which represent the product of biochemical rates and the time scale ( γ kc )) range from 0.4 to 1.5.

Parameters
The list of parameters used is given in Tables S1 and S2. The friction coefficients η and γ are set to be 100 nN.s/µm, from previous reports on embryo relaxation time obtained from laser ablation experiments, as in [Forgacs et al., 1998, Solon et al., 2009]. The stiffness of each spring, k c was set to 1 nN/µm per unit cell width to produce a relaxation time of 100 seconds, which falls in the range determined in [Solon et al., 2009]. In reality, cell stiffness could be a function of myosin content. The number of cells N, is estimated to be 30, from experimental evidence in [He, Wang, et al., 2010]. The rest length of the cell is assumed to be the average cell length from experimental data and is set to 6 µm, as in [He, Wang, et al., 2010]. Typical cell width, d is set to 5 µm. The stiffness of the angular springs k as is set to 4 nN.µm.
The pressure inside the egg chamber, P , the stiffness of the basement membrane, k and maximum contractile force F max are the variables in the system. Pressure is varied from 0.1 to 1 kPa, basement membrane stiffness is varied from 0.1 to 3 nN/µm per unit cell width. The range for maximum contractile force is chosen such that forces experienced by the cell are within the nN range as seen in experiments [Maruthamuthu et al., 2011, Gov 2009, Saez et al., 2010. It is varied from 10 to 60 nN per unit cell width.
The rates of activation and deactivation of Rho, ROCK and MLC are estimated to be as shown in Table S2 . The intrinsic rate of Rho GTP hydrolysis is on the order of 0.022 per minute (3x10 −4 per sec) which can be stimulated to several fold by GAP proteins [Zhang and Zheng, 1998]. The Rho activation rate used in this paper [TableS2] could be thought of as a rate upon stimulation by tension. We assume other rates to be on the same order. To represent cooperativity, Hill coefficients are commonly chosen to be greater than 2 [Jilkine et al., 2007, Winfree, 1980, Goldbeter, 1996, Ferrel Jr. et al. 2011], we choose a value of 10 in our model.

Additional Results
Activation of myosin as a function of egg chamber width Tension in the D-V direction of the egg chamber goes as internal pressure times the radius. As radius increases, increase in tension leads to activation of myosin. During stage 9 -10, as egg chamber grows in size, we see increased myosin accumulation due to increase in egg chamber width.( Figure S2) Effect of drugs on oscillation period and egg chamber radius.
Adding ionomycin, a drug that promotes contraction of actomyosin filaments, doubles the basal myosin intensity as well as reduces the egg chamber width by around 6% [He, Wang, et al., 2010]. In our model, increasing the rate of activation of MLC has the same effect. ( Figure S3) Treatment with various concentrations of Rock inhibitor Y-27632, decreased the myosin intensity but did not show huge variation in the oscillation period [He, Wang, et al., 2010]. Decreasing the rate of activation of ROCK shows non monotonous behaviour of the oscillation period and decrease in activated myosin fraction, consistent with experiments. (Figure S3) Analytic estimate of oscillation period from a single cell model For the single cell version of our model, the rate of change in cell length is given by External force acting on the cell is denoted by F ext . If F ext is made a linearly increasing function of time, we see that the system goes into a stable limit cycle within a range of forces ( Figure 2 in the main text). Here F ext acts as the bifurcation parameter, as F ext changes, the system changes from a steady state to an oscillatory limit cycle as seen in Figure 2. The dependence of the oscillation period on parameters such as cell stiffness k c , friction coefficient η and biochemical rates are studied using simulations. Since this full system of equations cannot be solved analytically, we compare period dependence from our simulations to the analytical solution for period obtained from a simpler single cell model at the bifurcation boundary.
A simple system to analytically solve is a two chemical component system, where we only have Rho and MLC. ROCK is assumed to be in steady state. The dependence of Rho on tension is simplified to be linear. The equations are as follows. where Performing linear stability analysis, the jacobian matrix for the above system of equations at steady state is The eigen values are then the solutions of the equation where Eq. (38) has either 1 or 3 negative roots given by Descartes rule of signs. For the three roots to have negative real parts, the Routh Hurwitz criterion [Hurwitz, 1895] is BC − D > 0. Hence, the bifurcation boundary is given by Solving equations 27, 28, 29 along with 42, gives the external force at the boundary as well as the steady state values at that force. At this force, the system moves from a limit cycle to steady state. At the bifurcation boundary, the analytic solution of the period is given by The bifurcation point in the full single cell model is found using simulations and the period is calculated at that external force. Unlike the simple model, this model has two bifurcation points, one at the onset of oscillations and one at its disappearance. To be consistent with the simple model, the bifurcation point where the limit cycle disappears is considered for period calculation. The value of force at this point is different compared to the value obtained in the simple model. Plots showing comparison of the full model simulation results with the analytical estimate are in Figures S4 and S5. Though the magnitude of the period is different in both cases, as they are different systems, the trends of period dependence on various factors seems to be the same at the bifurcation boundary.

Parameter study in the multi-cell model
In the multi-cell model, oscillation period follows a decreasing trend with increase in F max . On the other hand, the amplitude shows a consistent increase with increase in F max . Egg chamber width shows a decreasing trend with increase in F max , and as expected the mean myosin fraction shows an increase. The results shown in Figure S6 A and C are all at P = 0.4kPa. Countour plots of oscillation period,amplitude, egg chamber radius and mean activated myosin fraction are plotted as a function of P and F max in Figure S7. We further investigated the effect of the radius of cross section of eggchamber, the cell stiffness and angular spring stiffness between cell layers on the oscillation period. Angular spring constant did not play a role in affecting the period whereas a decrease in cell stiffness causes an increase in period (results shown at P = 0.3kPa, F max = 40pN per unit cell width in Figure S6B). At smaller eggchamber widths, higher contractile force (30 pN per unit cell width,0.2kPa) is required to cause oscillations, whereas at larger radii, higher pressures (10pN per unit cell width,1kPa) induce oscillations ( Figure  S6D).

Supplemental Figure Captions
• Figure S1: Raw data from experiments -Normalized myosin intensity plotted as a function of time in control and collagenase treated samples.
• Figure S2: Activation of myosin as a function of egg chamber radius -Activated myosin fraction increases as the egg chamber tension in the D-V direction increases. Increase in radius at the same internal pressure P implies increased tension.
• Figure S3: Pharmacological manipulation of egg chamber and its effects -Effect of adding ionomycin that increases the myosin intensity modeled as increasing the activation rate of MLC (A). Effect of ROCK inhibitor on oscillation period modeled as change in activation rate of ROCK(B). Parameters used are P = 0.1-0.3kPa and F max = 20-30nN per unit cell width.
• Figure S4: Single-cell model: Period dependence on mechanical parameters -The period at the bifurcation boundary decreases with increase in cell stiffness and maximum contractile force and increases with increase in the friction coefficient. Similar trends are observed in the full single cell model.
• Figure S5: Single-cell model: Period dependence on biochemical rates -The period at the bifurcation boundary decreases with increase in rates of activation and deactivation of Rho and MLC. The full single cell model simulation follows the same trends as the analytical estimate for period from a simpler model. Value of F max used in the single cell simulations is 10nN per unit cell width.
• Figure S6: Multi-cell model: Parameter study -Decrease in oscillation period (blue) and increase in amplitude (green) with increase in maximum contractile force (A). Decrease in egg chamber cross-section radius (blue) and increase in mean myosin intensity (green) with increase in maximum contractile force (C). Oscillation period decreases with increase in cell stiffness (B). Change from oscillations to steady state with increase in egg-chamber radius at low pressure and high contracile force (blue). Change from steady state to oscillations with increase in egg-chamber radius at high pressure and low maximum contracile force (red) (D).
• Figure S7: Multi-cell model: Contour plots of oscillation period, amplitude, egg chamber radius and mean activated myosin fraction as a function of maximum contractile force and internal pressure. Slight decrease in period (in the range 5-7 minutes), as a function of Fmax at all P above 0.3kPa(A). Increase in amplitude as a function of Fmax (B). Decrease in egg chamber radius as Fmax increases (C). Increase in myosin intensity as Fmax increases (D).
• Figure S8: Coupling of basement membrane to basal myosin in follicle cells. Using GFP-tagged endogenous talin and UAS-Paxillin driven by heat shock Gal4, we observed an enrichment of focal complex at the end of the actomyosin fibers, which suggests that basal myosin is mechanically coupled to the basement membrane.