Controlling organism size by regulating constituent cell numbers

Abstract

How living cells employ counting mechanisms to regulate their numbers or density is a long-standing problem in developmental biology that ties directly with organism or tissue size. Diverse cells types have been shown to regulate their numbers via secretion of factors in the extracellular space. These factors act as a proxy for the number of cells and function to reduce cellular proliferation rates creating a negative feedback. It is desirable that the production rate of such factors be kept as low as possible to minimize energy costs and detection by predators. Here we formulate a stochastic model of cell proliferation with feedback control via a secreted extracellular factor. Our results show that while low levels of feedback minimizes random fluctuations in cell numbers around a given set point, high levels of feedback amplify Poisson fluctuations in secreted-factor copy numbers. This trade-off results in an optimal feedback strength, and sets a fundamental limit to noise suppression in cell numbers with short-lived factors providing more efficient noise buffering. We further expand the model to consider external disturbances in key physiological parameters, such as, proliferation and factor synthesis rates. Intriguingly, while negative feedback effectively mitigates disturbances in the proliferation rate, it amplifies disturbances in the synthesis rate. In summary, these results provide unique insights into the functioning of feedback-based counting mechanisms, and apply to organisms ranging from unicellular prokaryotes and eukaryotes to human cells.

I. Introduction

In order to achieve physiological function various animals regulate their morphology [1]. For example, tissue size may be regulated by either controlling the sizes of individual cells, or the number of constituent cells [2]–[6]. How individual cells maintain size homeostasis has been extensively studied across organisms ranging from bacteria, animal and plant cells [7]–[15]. Interestingly, data reveals cell-autonomous control strategies that regulate cellular growth or timing of cell-cycle events to suppress aberrant deviations in cell size around an optimal size specific to that cell type [16]–[19]. In contrast, mechanisms that control cell numbers have been less understood.

One strategy to control cell numbers is for cells to secrete a molecule that accumulates in the extracellular space, and is sensed by other cells in the population [20]–[22]. Binding of this secreted factor to cell surface receptors activates signaling pathway that inhibit cell proliferation, thus creating a negative feedback loop (Fig. 1). Such feedback control via secreted factors has been reported in many organisms including Myxococcus xanthus [23], Vibrio fischeri [24], Bacillus subtilis [25], Dictyostelium discoideum [26], [27], multicellular animals [28], [29] and plants [30]. For illustration purposes, we show some published data on Dictyostelium discoideum in Fig. 1, where the proliferation rates of Dictyostelium discoideum cells decrease monotonically with increasing buildup of the secreted factor.

Fig. 1. Controlling cell numbers by feedback control through secreted extracellular factors.

Left. Schematic of a cell population, where each cell proliferates and secretes a molecule in the extracellular space. The buildup of secreted factors is sensed by other cells in the population. The factor functions to inhibit cell proliferation creating homeostasis in cell numbers. Right. Data showing proliferation rates of Dictyostelium discoideum cells with respect to the levels of two different secreted factors (AprA and Polyphosphate). While the inhibition of proliferation rate by AprA is gradual, that by Polyphosphate is relatively steep and occurs over a narrower range of factor levels. The error bars are the standard errors corresponding to the raw data, and lines represent Hill equations fitted to the data. We refer the reader to [33], [34] for further biological details and experimental procedures.

The feedback strategy illustrated in Fig. 1 creates an interesting tradeoff, where cells incur a cost to produce the factor, and would prefer to keep its synthesis to as low as possible [31], [32]. However, a side effect of low levels is shot noise or Poisson fluctuations in secreted factor copy numbers that are propagated to cell numbers. Here we study this tradeoff through a mathematical model that incorporates three different noise mechanisms: stochastic proliferation of cells; stochastic synthesis of the secreted factor from single cells; and external disturbances in model parameters. Our analysis reveals that depending on the source of noise, negative feedback can either buffer or amplify random fluctuations in cell numbers around a given set point. Moreover, when multiple noise sources are present, then an optimal feedback strength provides the most efficient noise buffering. This optimal feedback sets up a fundamental lower limit for minimizing fluctuations in cell numbers, and we systematically study how this limit scales with different parameters. We start by formally introducing the mathematical model followed by its stochastic analysis.

II. Stochastic model formulation

Let x(t) and z(t) denote the number of cells, and the number of secreted factors at time t, respectively. Cells are assumed to proliferate exponentially with a rate g (z), with g being a monotonically decreasing function as illustrated in Fig. 1. Moreover, cells are removed (or die) from the population at a rate γ_x. Finally, each cell synthesizes and secretes the factor at a rate k_z, and these secreted factors decay with a rate γ_z. The stochastic formation of this model is shown in Table 1 and consists of four probabilistic events that increase/decrease the population counts by one. The propensity functions in the last column determine how often the events occur. For example, the propensity function for the cell proliferation event is g (z) x which implies that the probability this event will occur in the next time interval (t, t + dt) is g (z) xdt, and whenever the event occurs the cell count increases by one.

TABLE I.

Stochastic events with their corresponding reset maps and propensity functions.

Event	Reset	Propensity function
Cell proliferation	x(t) ↦ x(t) + 1	g (z(t)) x(t)
Cell death	x(t) ↦ x(t) − 1	γxx(t)
Secreted factor synthesis	z(t) ↦ z(t) + 1	kzx(t)
Secreted factor degradation	z(t) ↦ z(t) − 1	γzz(t)

Throughout the paper we denote by 〈x(t)〉 as the mean value of the stochastic process x(t), and by $\overline{\langle x\rangle }={\text{lim}}_{t\to \infty }\langle x(t)\rangle$ its steady-state value. Based on the stochastic model in Table 1, the population averages evolve as

$$\frac{d\langle z\rangle }{dt}=k_z\langle x\rangle -{\gamma }_z\langle z\rangle$$ (1)

$$\frac{d\langle x\rangle }{dt}=\langle xg(z)\rangle -{\gamma }_x\langle x\rangle$$ (2)

[35], [36]. In the deterministic mean-field limit where

$$\langle xg(z)\rangle \approx \langle x\rangle g(\langle z\rangle )$$ (3)

we obtain the following approximated nonlinear system

$$\frac{d\langle z\rangle }{dt}=k_z\langle x\rangle -{\gamma }_z\langle z\rangle$$ (4)

$$\frac{d\langle x\rangle }{dt}=\langle x\rangle g(\langle z\rangle )-{\gamma }_x\langle x\rangle$$ (5)

that has a unique equilibrium defined by the solution to the equations

$$g (\overline{\langle z\rangle })={\gamma }_x,\overline{\langle x\rangle }=\frac{{\gamma }_z\overline{\langle z\rangle }}{k_z}$$ (6)

as long as g (0) > γ_x > g (∞). Having determined the mean equilibrium population counts, we next quantify the negative feedback strength by the dimensionless parameter

$$f:=-\frac{\overline{\langle z\rangle }{g}^{\prime }(\overline{\langle z\rangle })}{g(\overline{\langle z\rangle })}>0$$ (7)

that is essentially the log sensitivity of g at $z=\overline{\langle z\rangle }$. A Hill type equation given by

$$g(z)=(g(0)-g(\infty ))\frac{K^n}{K^n+z^n}+g(\infty )$$ (8)

captures the functional dependence of the proliferation rate on the secreted factor level adequately. Here n and K are the Hill coefficient and the half-saturation constant respectively. Using this assumption, (7) becomes

$$f=\frac{n\left({\gamma }_x-g(\infty )\right)\left(g(0)-{\gamma }_x\right)}{{\gamma }_x(g(0)-g(\infty ))}.$$ (9)

Note that while the feedback strength is proportional to the Hill coefficient of the proliferation rate, it also depends on the degradation rate of the secreted factor which is part of the feedback loop. Local stability analysis of the equilibrium point yields the following eigenvalues of the linearized system

$$\frac{-{\gamma }_z\pm \sqrt{{\gamma }_z^2-4f{\gamma }_x{\gamma }_z}}{2}$$ (10)

and shows that the equilibrium of the nonlinear system (4)–(5) is stable for f > 0. Note that sufficiently strong negative feedback (f > γ_z/4γ_x) results in complex eigenvalues with negative real parts, and hence, damped oscillatory dynamics.

III. Optimal feedback strength for regulating cell numbers

Stochastic simulations of the feedback system reveal an intriguing feature - while low levels of feedback strength attenuate random fluctuations in cell numbers, a strong negative feedback amplifies fluctuations (Fig. 2). We further investigate this effect by developing approximate analytical formulas for the noise in cell numbers, where noise is quantified by the steady-state coefficient of variation of x(t). A well-known problem that often arises when dealing with nonlinear stochastic systems is unclosed moment dynamics - time evolution of lower order moments depends on higher order moments [38], [39]. While a number of closure methods have been developed to tackle this issue [38]–[40], we circumvent this problem by exploiting the Linear Noise Approximation (LNA) [41], [42]. Assuming small fluctuations in x(t) and z(t) around their respective steady-state means, LNA works by linearizing nonlinear propensity functions, as in

$$xg(z)\approx g(\overline{\langle z\rangle })\left(x-f\overline{\langle x\rangle }\frac{z-\overline{\langle z\rangle }}{\overline{\langle \mathit{z}\rangle }}\right),$$ (11)

and with this approximation all propensity functions in Table 1 are now linear functions of the state space. Time evolution of statistical moments is obtained using the following result from our prior work [36], [39]: the time derivative of the expected value of an arbitrary function ψ(x, z) is given by

$$\frac{d\langle \psi (x,z)\rangle }{dt}=\langle k_{z}x[\psi (x,z+1)-\psi (x,z)]\rangle +\langle g(\overline{\langle z\rangle })\overline{\langle x\rangle }\left(\frac{x}{\overline{\langle x\rangle }}-f\frac{z-\overline{\langle z\rangle }}{\overline{\langle \mathit{z}\rangle }}\right)[\psi (x+1,z)-\psi (x,z)]\rangle +\langle {\gamma }_{x}x[\psi (x-1,z)-\psi (x,z)]\rangle +\langle {\gamma }_{z}z[\psi (x,z-1)-\psi (x,z)]\rangle .$$ (12)

Fig. 2. Noise in cell numbers is minimized at an intermediate level of negative feedback.

A. Stochastic simulation of the model presented in Table 1 using [37] for different feedback strengths (f = 0.01 (Low), f = 0.6 (Optimal), f = 10 (High)). For a given mean number of cells, the spread in cell numbers first reduces, and then increases with increasing feedback strength. B. Plotting the total noise, and the different noise components in (18) as a function of f. While the noise contribution from stochastic proliferation reduces, the contribution from stochastic secreted-factor synthesis increases creating a U-shape profile of the total noise. Parameters chosen for all the above subfigures were: γ_z = 3γ_x = 3day⁻¹, $\overline{\langle z\rangle }=100$, $\overline{\langle x\rangle }=150$. The functional form of the proliferation rate obtained by substituting the above parameters in (6), (8), (9) was $g(z)=2\frac{{\overline{\langle z\rangle }}^{2f}}{z^{2f}+{\overline{\langle z\rangle }}^{2f}}da{y}^{-1}$.

Substituting appropriate monomials for ψ(x, z) above yields the following system of differential equations for all the first and second order moments of x(t) and z(t)

$$\frac{d\langle x\rangle }{dt}=g(\overline{\langle z\rangle })\overline{\langle x\rangle }\left(\frac{x}{\overline{\langle x\rangle }}-f\frac{\langle z\rangle -\overline{\langle z\rangle }}{\overline{\langle \mathit{z}\rangle }}\right)-{\gamma }_x\langle x\rangle$$ (13)

$$\frac{d\langle z\rangle }{dt}=k_z\langle x\rangle -{\gamma }_z\langle z\rangle$$ (14)

$$\frac{d\langle xz\rangle }{dt}=g(\overline{\langle z\rangle })\overline{\langle x\rangle }\left(\frac{\langle xz\rangle }{\overline{\langle x\rangle }}-f\frac{\langle z^2\rangle -\overline{\langle z\rangle }\langle z\rangle }{\overline{\langle z\rangle }}\right)+k_z\langle x^2\rangle -\left({\gamma }_x+{\gamma }_z\right)\langle xz\rangle$$ (15)

$$\frac{d\langle x^2\rangle }{dt}={\gamma }_x\left(\langle x\rangle -2\langle x^2\rangle \right)+g(\overline{\langle z\rangle })\overline{\langle x\rangle }\left(\frac{x}{\overline{\langle x\rangle }}-f\frac{\langle z\rangle -\overline{\langle z\rangle }}{\overline{\langle z\rangle }}\right)+2g(\overline{\langle z\rangle })\overline{\langle x\rangle }\left(\frac{\langle x^2\rangle }{\overline{\langle \mathit{x}\rangle }}-f\frac{\langle xz\rangle -\langle x\rangle \overline{\langle z\rangle }}{\overline{\langle \mathit{z}\rangle }}\right)$$ (16)

$$\frac{d\langle z^2\rangle }{dt}={\gamma }_z\left(\langle z\rangle -2\langle z^2\rangle \right)+k_z\langle x\rangle +2k_z\langle xz\rangle .$$ (17)

Solving the moment equations at steady state we obtain the following noise (coefficient of variation squared) in cell numbers

$$C{V}_x^2=\underset{\text{Noise from secreted factor}}{\underbrace{\frac{{\gamma }_{x}f}{{\gamma }_z\overline{\langle z\rangle }}}}+\underset{\text{Noise from stochastic proliferation}}{\underbrace{\frac{{\gamma }_z+{\gamma }_{x}f}{{\gamma }_{z}f\overline{\langle x\rangle }}}}$$ (18)

which can be decomposed into two terms. The first term represents the noise contribution from Poisson fluctuations in z(t) (noise from secreted factor) and the other term is the contribution from stochastic proliferation/death of cells. Note that CV_x decreases as the secreted factor half-life decreases, which makes intuitive sense as then the regulator tracks the cell numbers more faithfully. Interestingly, the first term is amplified, and second term is attenuated with increasing feedback strength f (Fig. 2). These opposing effects result in $C{V}_x^2$ being minimized at an optimal feedback strength

$$f=\frac{\sqrt{{\gamma }_z\overline{\langle z\rangle }}}{\sqrt{{\gamma }_x\overline{\langle x\rangle }}}.$$ (19)

Replacing (19) in (18) and using (3) yields the fundamental limit of noise suppression in cell numbers as

$$C{V}_x^2=\frac{\frac{{\gamma }_x}{{\gamma }_z}+\frac{2\sqrt{{\gamma }_x}}{\sqrt{k_z}}}{\overline{\langle x\rangle }}.$$ (20)

While inverse scaling of $C{V}_x^2$ with respect to the mean $\overline{\langle x\rangle }$ is reminiscent of Poisson-like fluctuations, the numerator value can be made small by having a short-lived secreted factor (high γ_z) and a high factor synthesis rate k_z. It is interesting to point out that the ratio N = k_z/γ_x is the average number of secreted factors made by an individual cell in its lifespan, and the fundamental noise limit scales as $1/\sqrt{N}$, but the scaling is 1/γ_z with respect to the factor decay rate.

IV. Incorporating external disturbances in physiological parameters

Our analysis up till now assume constant model parameters, and we now expand the model to allow for external disturbances that transform parameters into stochastic processes.

A. External disturbance in the proliferation rate

Data shows that the cellular growth rate can randomly fluctuate with some memory over many generations [43]–[47], and its not hard to imagine these features carrying over to the cell proliferation rate. Motivated by these findings we assume that the proliferation rate g (z, y) is affected by an external factor y whose stochastic dynamics is modeled as an Ornstein-Uhlenbeck process

$$dy(t)={\gamma }_y(\overline{\langle y\rangle }-y(t))dt+{\sigma }_{y}dw(t).$$ (21)

Here, $\overline{\langle y\rangle }$ is the mean level of external disturbance, w(t) is the Wiener process, σ_y and γ_y are parameters that represent the strength of noise and the time-scale of fluctuations in y, respectively. Linearizing the proliferation rate

$$\begin{gathered} xg(z,y)\approx \overline{\langle x\rangle }g(\overline{\langle z\rangle },\overline{\langle y\rangle })\left(\frac{x}{\overline{\langle x\rangle }}-f\frac{z-\overline{\langle z\rangle }}{\overline{\langle \mathit{z}\rangle }}+S_x\frac{y-\overline{\langle y\rangle }}{\overline{\langle \mathit{y}\rangle }}\right), \\ {S}_x:={\frac{\overline{\langle y\rangle }}{g(\overline{\langle z\rangle },\overline{\langle y\rangle })}\frac{\partial g(\overline{\langle z\rangle },y)}{\partial y}|}_{y=\overline{\langle y\rangle }} \end{gathered}$$ (22)

where S_x is the log sensitivity of the proliferation rate to the external disturbance. With this approximations we again repeat the analysis of obtaining moment equations and solving it to get noise in cell numbers. We write the moment dynamics of an arbitrary function ψ(x, y, z) of the states of the system using the extended generator L of stochastic hybrid systems [35], [36], [39]

$$\frac{d\langle \psi (x,y,z)\rangle }{dt}=\langle (L\psi )(x,y,z)\rangle .$$ (23)

For the system in consideration, the extended generator is given by

$$\begin{gathered} (L\psi )(x,y,z):=\frac{\partial \psi (x,y,z)}{\partial y}{\gamma }_y(\overline{\langle y\rangle }-y)+\frac{1}{2}\left(\frac{{\partial }^2\psi (x,y,z)}{\partial y^2}{\sigma }_y^2\right) \\ +k_{z}x[\psi (x,y,z+1)-\psi (x,y,z)] \\ +g(\overline{\langle z\rangle },\overline{\langle y\rangle })\overline{\langle x\rangle }\left(\frac{x}{\overline{\langle \mathit{x}\rangle }}-f\frac{z-\overline{\langle z\rangle }}{\overline{\langle z\rangle }}+S_x\frac{y-\overline{\langle y\rangle }}{\overline{\langle y\rangle }}\right) \\ \times [\psi (x+1,y,z)-\psi (x,y,z)]+{\gamma }_{x}x[\psi (x-1,y,z)-\psi (x,y,z)] \\ +{\gamma }_{z}z[\psi (x,y,z-1)-\psi (x,y,z)]. \end{gathered}$$ (24)

Substituting appropriately monomials for ψ(x, y, z) we obtain the following moment dynamics

$$\frac{d\langle x\rangle }{dt}=g(\overline{\langle z\rangle },\overline{\langle y\rangle })\overline{\langle x\rangle }\left(\frac{\langle x\rangle }{\overline{\langle x\rangle }}-f\frac{\langle z\rangle -\overline{\langle z\rangle }}{\overline{\langle z\rangle }}+S_x\frac{\langle y\rangle -\overline{\langle y\rangle }}{\overline{\langle y\rangle }}\right)-{\gamma }_x\langle x\rangle$$ (25)

$$\frac{d\langle z\rangle }{dt}=k_z\langle x\rangle -{\gamma }_z\langle z\rangle$$ (26)

$$\frac{d\langle y\rangle }{dt}={\gamma }_y(\overline{\langle y\rangle }-\langle y\rangle )$$ (27)

$$\frac{d\langle xz\rangle }{dt}=g(\overline{\langle z\rangle },\overline{\langle y\rangle })\overline{\langle x\rangle }\left(\frac{\langle xz\rangle }{\overline{\langle x\rangle }}-f\frac{\langle z^2\rangle -\overline{\langle z\rangle }\langle z\rangle }{\overline{\langle z\rangle }}+S_x\frac{\langle yz\rangle -\overline{\langle y\rangle }\langle z\rangle }{\overline{\langle y\rangle }}\right)+k_z\langle x^2\rangle -\left({\gamma }_x+{\gamma }_z\right)\langle xz\rangle$$ (28)

$$\frac{d\langle xy\rangle }{dt}=g(\overline{\langle z\rangle },\overline{\langle y\rangle })\overline{\langle x\rangle }\left(\frac{\langle xy\rangle }{\langle x\rangle }-f\frac{\langle yz\rangle -\overline{\langle z\rangle }\langle y\rangle }{\overline{\langle z\rangle }}+S_x\frac{\langle y^2\rangle -\overline{\langle y\rangle }\langle y\rangle }{\overline{\langle y\rangle }}\right)+{\gamma }_y(\langle x\rangle \overline{\langle y\rangle }-\langle xy\rangle )-{\gamma }_x\langle xy\rangle$$ (29)

$$\frac{d\langle yz\rangle }{dt}={\gamma }_y(\overline{\langle y\rangle }\langle z\rangle -\langle yz\rangle )+k_z\langle xy\rangle -{\gamma }_z\langle yz\rangle$$ (30)

$$\frac{d\langle x^2\rangle }{dt}={\gamma }_x\left(\langle x\rangle -2\langle x^2\rangle \right)+g(\overline{\langle z\rangle },\overline{\langle y\rangle })\overline{\langle x\rangle }\left(\frac{\langle x\rangle }{\overline{\langle x\rangle }}-f\frac{\langle z\rangle -\overline{\langle z\rangle }}{\overline{\langle z\rangle }}+S_x\frac{\langle y\rangle -\overline{\langle y\rangle }}{\overline{\langle y\rangle }}\right)+2g(\overline{\langle z\rangle },\overline{\langle y\rangle })\overline{\langle x\rangle }\left(\frac{\langle x^2\rangle }{\overline{\langle x\rangle }}-f\frac{\langle xz\rangle -\langle x\rangle \overline{\langle z\rangle }}{\overline{\langle z\rangle }}+S_x\frac{\langle xy\rangle -\langle x\rangle \overline{\langle y\rangle }}{\overline{\langle y\rangle }}\right)$$ (31)

$$\frac{d\langle y^2\rangle }{dt}={\gamma }_y\left(2\overline{\langle y\rangle }\langle y\rangle -2\langle y^2\rangle \right)+{\sigma }_y^2$$ (32)

$$\frac{d\langle z^2\rangle }{dt}={\gamma }_z\left(\langle z\rangle -2\langle z^2\rangle \right)+k_z\langle x\rangle +2k_z\langle xz\rangle .$$ (33)

Steady-state analysis of this system of differential equations yields the following noise in cell numbers

$$C{V}_x^2=\underset{\text{Noise from secreted factor}}{\underbrace{\frac{{\gamma }_{x}f}{{\gamma }_z\overline{\langle z\rangle }}}}+\underset{\text{Noise from stochastic proliferation}}{\underbrace{\frac{{\gamma }_z+{\gamma }_{x}f}{{\gamma }_{z}f\overline{\langle x\rangle }}}}+\underset{\text{Noise from external disturbance}}{\underbrace{\frac{C{V}_y^2{\gamma }_x{S}_x^2\left({\gamma }_z^2+{\gamma }_z{\gamma }_y+f{\gamma }_x{\gamma }_y\right)}{{\gamma }_{z}f\left({\gamma }_x{\gamma }_{z}f+{\gamma }_y\left({\gamma }_z+{\gamma }_y\right)\right)}}}.$$ (34)

Comparing with (18) we see an additional third term that represents the contribution from the external disturbance, and this term monotonically decreases to zero as f → ∞ (Fig. 3). Assuming a short-lived secreted factor γ_z ≫γ_x, γ_y, (34) reduces to

$$C{V}_x^2=\underset{\text{Noise from secreted fector }}{\underbrace{\frac{{\gamma }_{x}f}{{\gamma }_z\overline{\langle z\rangle }}}}+\underset{\text{Noise from stochastic proliferation}}{\underbrace{\frac{1}{f\overline{\langle x\rangle }}}}+\underset{\text{Noise from external disturbance}}{\underbrace{\frac{C{V}_y^2{\gamma }_x{S}_x^2}{f\left({\gamma }_{x}f+{\gamma }_y\right)}}}.$$ (35)

Fig. 3. Depending on the source of noise, random fluctuations in cell numbers can be amplified or buffered with increasing negative feedback strength.

Plots for the noise components shown in (34) and (37) as a function of f. While adding feedback buffers noise from stochastic proliferation and disturbances in the proliferation rate, it amplifies any noise associated with the secreted factor. The latter includes shot noise or Poisson fluctuation in secreted factor copy numbers, and external disturbances to the synthesis rate. Parameters chosen were: $C{V}_y^2=0.05$, ${\gamma }_z=5{\gamma }_x$, ${\gamma }_y={\gamma }_x/5,\overline{\langle x\rangle }=200$, $\overline{\langle z\rangle }=80$.

While the noise contribution from stochastic proliferation scales as f⁻¹, depending on the relative values of γ_x, γ_y the contribution from external disturbance scales between f⁻¹ and f⁻². This emphasizes a key point that feedback regulation is more efficient in suppressing noise arising from disturbances in proliferation rate as compared to inherent stochasticity from small cell numbers. We next consider external disturbance in the synthesis rate k_z while reverting the proliferation rate back to being a constant.

B. External disturbance in the synthesis rate

The activity of the feedback secreted factors is usually mediated by secondary molecules [48]. This introduces a disturbance which is external to the feedback system. To model this feature the external disturbance, which follows the dynamics in (21), affects the synthesis rate of the secreted factor as

$$\begin{gathered} k_z(y)x=k_z(\overline{\langle y\rangle })\overline{\langle x\rangle }\left(\frac{x}{\overline{\langle x\rangle }}+S_z\frac{z-\overline{\langle z\rangle }}{\overline{\langle z\rangle }}\right), \\ {S}_z:=\frac{\overline{\langle y\rangle }}{k_z(\overline{\langle y\rangle })}{\frac{\partial k_z(y)}{\partial y}|}_{y=\overline{\langle y\rangle }}. \end{gathered}$$ (36)

Here S_z is the log sensitivity of the secreted factor synthesis rate to the external disturbance. By performing an analysis similar to the previous section we obtain the following noise in cell numbers

$$C{V}_x^2=\underset{\text{Noise from secreted factor}}{\underbrace{\frac{{\gamma }_{x}f}{{\gamma }_z\overline{\langle z\rangle }}}}+\underset{\text{Noise from stochastic proliferation}}{\underbrace{\frac{{\gamma }_z+f{\gamma }_x}{f{\gamma }_z\overline{\langle x\rangle }}}}+\underset{\text{ Noise from external disturbance }}{\underbrace{\frac{f{S}_z^{2}C{V}_y^2{\gamma }_x\left({\gamma }_z+{\gamma }_y\right)}{f{\gamma }_x{\gamma }_z+{\gamma }_z{\gamma }_y+{\gamma }_y^2}}}$$ (37)

and now the contribution from external disturbance increases with increasing feedback strength f (Fig. 3). This is intuitive, as the disturbance acts through the secreted factor, and adding more feedback only functions to propagate this disturbance to affect cell numbers.

V. Conclusion

Regulation of cell number is critical in the functioning of the systems they constitute. For example in Myxococcus xanthus the formation of fruiting bodies only occurs in presence of precise cell densities and starvation. Further in Dictyostelium, during starvation, the cells aggregate to form spores in a fruiting body which is supported by a stalk [49]. If number of spores in the fruiting body is too small, it will be too close to the ground and cause inefficient spore dispersal. However, if the number of spores is too large then it falls over, possibly ruining a chance of germination of the spores when nutrients become available. In this paper we study the maintenance of precise cell numbers conferred by negative feedback of secreted factors. We show that for low feedback strength of the secreted factor, the noise in cell numbers will be very high (Fig. 2). This leads to cell groups with high cell numbers, as observed experimentally in Dictyoselium discoideum with loss of regulation [34], [48].

Note that in resource deficient environments, producing low levels of secreted factors would be desirable albeit with significant noise in the secreted factor. We show that there exists a trade-off between the increase in noise due to the secreted factor and the reduction in noise due to stochastic proliferation. This leads to an optimal feedback feedback strength given by (19) which minimizes noise in cell number (20). Further external disturbance in proliferation rate increases noise in cell number although it can be buffered by the secreted factor to arbitrarily low levels. Note that with increase in the time-scale of the external disturbance we see that noise in cell number increases. This is due to the increased memory in the external fluctuations and consequently proliferation rate through time.

In conclusion our study uses a simple feedback model to study the regulation of proliferation of cells. The results may guide future experimental studies to design regulatory strategies to minimize cell number variation and consequently tissues/organ size to influence morphology and function of animals. This study can be used in further studies to give an evolutionary rationale to the existence of the negative feedback structure similar to that in gene auto-regulatory loops [50]–[52]. The study can also be extended to cases where the secreted factor is interlinked between two types of cells in the same environment [53].