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High Cooperativity in Negative Feedback can Amplify Noisy Gene Expression

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Abstract

Burst-like synthesis of protein is a significant source of cell-to-cell variability in protein levels. Negative feedback is a common example of a regulatory mechanism by which such stochasticity can be controlled. Here we consider a specific kind of negative feedback, which makes bursts smaller in the excess of protein. Increasing the strength of the feedback may lead to dramatically different outcomes depending on a key parameter, the noise load, which is defined as the squared coefficient of variation the protein exhibits in the absence of feedback. Combining stochastic simulation with asymptotic analysis, we identify a critical value of noise load: for noise loads smaller than critical, the coefficient of variation remains bounded with increasing feedback strength; contrastingly, if the noise load is larger than critical, the coefficient of variation diverges to infinity in the limit of ever greater feedback strengths. Interestingly, feedbacks with lower cooperativities have higher critical noise loads, suggesting that they can be preferable for controlling noisy proteins.

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Notes

  1. Biologically, the mean size of a translational burst is equal to the ratio of the translation rate \(k_\mathrm {p}\) to the mRNA decay rate \(\gamma _\mathrm {m}\) (Friedman et al. 2006). We assume that the degradation of mRNA is catalysed by the protein, such as in

    $$\begin{aligned} \text {mRNA} \xrightarrow {\gamma _0} \emptyset ,\quad \text {mRNA} + H \times \text {protein} \xrightarrow {\gamma _1} H \times \text {protein}, \end{aligned}$$

    where \(\gamma _0\) is the natural decay rate and \(\gamma _1\) measures the efficiency of the catalysis. The total rate of mRNA decay is then given by \(\gamma _\mathrm {m} = \gamma _0 + \gamma _1 x^H\), and the mean burst size is equal to \(k_\mathrm {p}/(\gamma _0 + \gamma _1 x^H)\), which can be turned into (6) by a suitable choice of unit concentration. The above reaction channels form the backbone of a full discrete chemical kinetics model, which is examined and compared to our present continuous formulation in “Appendix A”.

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Acknowledgements

We thank an anonymous referee for useful comments and important insights, in particular those leading to the analysis of “Appendix A”.

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Correspondence to Pavol Bokes.

Additional information

PB is supported by the Slovak Research and Development Agency Grant APVV-14-0378 and also by the VEGA Grant 1/0319/15. YTL thanks for the support from the Center for Nonlinear Studies, Los Alamos National Laboratory and the Engineering and Physical Sciences Research Council EPSRC (UK) (Grant No. EP/K037145/1). AS is supported by the National Science Foundation Grant DMS-1312926.

Appendices

Appendix A: Discrete Model

Here we compare the results of the Main Text to a fine-grained chemical kinetics model consisting of three discrete species (mRNA, PreProtein, and Protein) that are subject to reaction channels

$$\begin{aligned} \begin{aligned}&\emptyset \xrightarrow {\frac{1}{\varepsilon }} \text {mRNA},&\text {mRNA} \xrightarrow {\frac{1}{\delta }} \emptyset ,&\text {mRNA} + \text {Protein} \xrightarrow {\frac{\text {Protein}}{\delta \Omega K}} \text {Protein},\\&\text {Protein} \xrightarrow {1} \emptyset ,&\text {PreProtein} \xrightarrow {\frac{1}{\lambda }} \text {Protein},&\text {mRNA} \xrightarrow {\frac{\varepsilon \Omega }{\delta }} \text {mRNA} + \text {PreProtein}. \end{aligned} \end{aligned}$$
(41)

These six reactions represent the following processes: (i) mRNA synthesis (burst initiation); (ii) natural mRNA decay; (iii) mRNA decay catalysed by the protein; (iv) protein decay (v) protein maturation; (vi) translation. We restrict ourselves, for simplicity, to non-cooperative feedback in the third reaction, i.e. \(H=1\) in the sense of the Main Text. The reaction constants in (41) are parameterised in a manner that relates the discrete model to its continuous counterpart of the Main Text. In addition to the familiar parameters \(\varepsilon \) (noise load) and K (critical concentration), three new parameters appear: the system-size parameter \(\Omega \) gives the number of protein molecules in a unit of concentration; \(\delta \) and \(\lambda \) give the mRNA lifetime and the mean protein maturation delay, respectively, in units of the protein lifetime.

While a systematic asymptotic analysis of (41) is beyond the scope of this paper, we expect that in the limit of \(\Omega \rightarrow \infty \), the discrete model reduces to a piecewise deterministic Markov process (PDMP), with one discrete species (mRNA) and two continuous species (pre-protein and mature protein). In the limit of \(\delta \rightarrow 0\), whilst keeping \(\lambda \) finite positive, mRNA is adiabatically eliminated from the model, and pre-protein is produced in instantaneous bursts. The size of a burst is drawn from a geometric distribution, the mean of which decreases with the amount of mature protein in the system. If one additionally takes \(\lambda \rightarrow 0\) after the limit of \(\delta \rightarrow 0\) has already been taken, then the system features an infinitesimal delay in the sense of Sect. 7. Combining the above limits of \(\Omega \rightarrow \infty \) and \(\delta \rightarrow 0\) before \(\lambda \rightarrow 0\), one expects to arrive at the continuous-state bursting model with infinitesimal delay, on which we focused throughout the Main Text.

Fig. 9
figure 9

(Color figure online) Noise (CV\(^2/\varepsilon \)) as function of feedback strength (\(K^{-1}\)). We use the continuous-state drift-jump model with infinitesimal delay (solid black) or without infinitesimal delay (dashed black), and the discrete model (41) (coloured circles). Shaded bands indicate \(95\%\) confidence intervals

The species means and standard deviations were calculated using Stochpy’s (Maarleveld et al. 2013) implementation of Gillespie’s direct method (Gillespie 1976). We skipped the first 30 units of time to reach stationarity and used the next \(10^6\) iterations to estimate the moments. The moments were used, in particular, to estimate the protein coefficient of variation. The procedure was repeated ten times to improve precision by additional ensemble averaging and to obtain approximate confidence intervals (assuming asymptotic normality).

In the present simulation, we fixed \(\varepsilon =1\), \(H=1\), and varied K between \(10^{-2}\) and \(10^2\). The noise load \(\varepsilon =1\) exceeds the critical value \(\varepsilon _\mathrm{c}=1/2H = 0.5\). Comparing the results of the full discrete and continuous bursting simulation in Fig. 9, we are able to draw the following conclusions:

  • Finiteness of \(1/\lambda \) reduces the noise. It takes \(\lambda \) units of time, on average, for a pre-protein to mature. The intrinsic variability of maturation times attenuates the effective burst size and reduces protein noise. Finiteness of \(1/\lambda \) is responsible, in particular, for the relative decrease of noise in weak feedback conditions (Fig. 9, blue and orange markers). Under strong-feedback conditions, other effects, which are listed below, dominate.

  • Finiteness of \(\lambda /\delta \) also reduces the noise. Large values of \(\lambda /\delta \) mean that few proteins will have matured over the duration of a single burst. Nevertheless, if feedback is strong, these few proteins may suffice to stop the burst and ameliorate noise (Fig. 9, blue markers). An additional increase in \(\lambda /\delta \) is required to offset the effect (Fig. 9, orange markers).

  • Finiteness of \(\Omega \) increases the noise. A unit of concentration comprises \(\Omega \) copies of protein. Strong feedback decreases the mean protein concentration, making the low-copy-number Poissonian noise increasingly more important. We attribute to this effect, in particular, the excess noise exhibited by the discrete model under strong-feedback conditions (Fig. 9, orange and green markers).

Aside from exhibiting the aforementioned additional effects, the discrete model reflects the essential features of the continuous bursting descriptions of the Main Text, in particular with respect to the interaction between the bursting and maturation timescales, and their effect on protein noise.

Appendix B: Estimating Protein Moments from Simulations

We can estimate the n-th steady-state moment by the time average \(\frac{1}{T}\int _0^T x^n(t) \mathrm{d}t\), where \(T\gg 1\), of a sample trajectory x(t) generated by Algorithm 1 and raised to the power of n. Since x(t) is piecewise exponential, we have

$$\begin{aligned} \frac{1}{T}\int _0^T x^n(t) \mathrm{d}t = \frac{1}{T}\sum _{i=0}^{N-1} \int _0^{\tau _i} x_i^n \mathrm{e}^{-n t} \mathrm{d}t = \frac{1}{n T}\sum _{i=0}^{N-1} x_i^n (1 - \mathrm{e}^{-\tau _i n}), \end{aligned}$$
(42)

where \(x_i\), \(i\ge 1\), is the protein concentration immediately after the i-th burst, \(x_0\) is the initial protein concentration, \(\tau _i\), \(i\ge 1\), is the waiting time from the i-th burst until the \((i+1)\)-th burst, \(\tau _0\) is the waiting time from the initial time until the first burst, and \(T = \sum _{i=0}^{N-1} \tau _i\), where N is a large integer. By Algorithm 1, the values of \(x_i\) and \(\tau _i\) are obtained by

$$\begin{aligned} x_{i+1} = x_i \mathrm{e}^{-\tau _i} - \frac{\varepsilon \mathrm{ln}\tilde{u}_i}{1 + ( x_i \mathrm{e}^{-\tau _i}/K)^H},\quad x_0=0,\quad \tau _i = -\varepsilon \mathrm{ln} u_i,\quad i=0,1,\ldots ,\qquad \end{aligned}$$
(43)

where \(u_i\) and \(\tilde{u}_i\), \(i=0,1,\ldots \), are random variates drawn independently of each other from the uniform distribution in the unit interval.

Inserting \(T\approx \varepsilon N\), which holds by the law of large numbers, into (42), and shifting the time frame to reduce the effect of the transient behaviour, we arrive at an estimate

$$\begin{aligned} \widehat{\langle x^n \rangle } = \frac{1}{\varepsilon n N}\sum _{i=M}^{N+M-1} x_i^n (1 - \mathrm{e}^{-\tau _i n}), \end{aligned}$$
(44)

where \(x_i\) and \(\tau _i\) are given by (43). We used (44) with \(N=10^9\) and \(M=10^7\) to estimate the theoretical value of \(\langle x^n\rangle \) in Figs. 3, 4, 5 and 8. In the strong-feedback limit (Figs. 6, 7), ensemble averaging across a large number of sample paths was needed for more precise measurements of the power-law exponents.

Appendix C: Strong-Feedback Asymptotics in Critical Cases

The dominant contribution to the n-th moment of a solution p(x) to (20) comes exclusively from the inner \(O(K^{H/(H+1)})\) concentration scale if \(n<1/\varepsilon H\) or from the outer O(1) concentration scale if \(n>1/\varepsilon H\). In the borderline case of \(n=1/\varepsilon H\), which we explore in this Appendix, the inner and outer regions both contribute to the leading-order behaviour of the n-th moment. The individual contributions can be identified by splitting the range of integration (Hinch 1991),

$$\begin{aligned} m_n = \int _0^\infty p(x) x^n \mathrm{d}x = \int _0^\delta p(x) x^n \mathrm{d}x + \int _\delta ^\infty p(x) x^n \mathrm{d}x, \end{aligned}$$
(45)

where \(\delta \) is a value taken from an intermediate scale (\(K^{H/(H+1)}\ll \delta \ll 1\)).

In the first integral on the right-hand side of (45), we substitute \(x=K^{H/(H+1)}y\) and, neglecting the slowly decaying exponential in (21), we obtain

$$\begin{aligned} \int _0^\delta p(x) x^n \mathrm{d}x \sim K^\frac{(n+1)H}{H+1}\int _0^{K^{-\frac{H}{H+1}}\delta } P(y) y^n \mathrm{d}y. \end{aligned}$$
(46)

Since \(\delta \gg K^{H/(H+1)}\), the upper integration limit of the integral on the right-hand side of (46) is large. The integral itself diverges to infinity, since \(n=1/\varepsilon H\) together with (23b) imply that \(P(y) y^n \sim \eta y^{-1}\) for \(y\gg 1\). Extricating the divergent part from the integral in (46), we obtain an asymptotic approximation

$$\begin{aligned} \int _0^{K^{-\frac{H}{H+1}}\delta } P(y) y^n \mathrm{d}y&= \int _0^{K^{-\frac{H}{H+1}}\delta } \left( P(y) y^n - \frac{\eta }{y+1}\right) \mathrm{d}y + \eta \int _0^{K^{-\frac{H}{H+1}}\delta } \frac{\mathrm{d}y}{y+1}\nonumber \\&\sim \int _0^\infty \left( P(y) y^n - \frac{\eta }{y+1}\right) \mathrm{d}y - \frac{ \eta H }{H+1}\mathrm{ln}K + \eta \mathrm{ln}\delta . \end{aligned}$$
(47)

In the second integral on the right-hand side of (45), we approximate the integrand with (24b), finding that

$$\begin{aligned} \int _\delta ^\infty p(x) x^n&\mathrm{d}x \sim \eta K^\frac{(n+1)H}{H+1} E_1\left( \frac{\delta }{\varepsilon }\right) \sim \eta K^\frac{(n+1)H}{H+1} ( - \mathrm{ln}\delta + \mathrm{ln}\varepsilon - \gamma ), \end{aligned}$$
(48)

where \(E_1(t)\) is the exponential integral and \(\gamma =0.577\ldots \) is the Euler–Mascheroni constant (Abramowitz and Stegun 1972).

Combining (45)–(48), we find that the n-the moment can be expanded into

$$\begin{aligned} m_n \sim K^\frac{(n+1)H}{H+1} \left( \int _0^\infty \left( P(y) y^n - \frac{\eta }{y+1}\right) \mathrm{d}y - \frac{ \eta H }{H+1}\mathrm{ln}K + \eta ( \mathrm{ln}\varepsilon - \gamma )\right) . \end{aligned}$$
(49)

Although the logarithmic term in (49) asymptotically dominates, as K tends to zero, the neighbouring constant terms, in practice the magnitudes of the logarithm and the constant terms are similar so the latter cannot be neglected.

As a particular application of the expansion (49), we evaluate the small-K behaviour of the protein coefficient of variation subject to a critical noise load \(\varepsilon ={1}/{2H}\). The leading-order approximations to \(m_0\) and \(m_1\) involve contributions from the inner \(O(K^{H/(H+1)})\) scale only and are given by (31). On the other hand, the leading-order approximation to \(m_2\) combines contributions from both inner and outer scales and is given by (49). For the coefficient of variation we obtain

$$\begin{aligned} \mathrm{CV}^2&= \frac{m_2 m_0}{m_1^2} - 1 \sim \frac{\nu _0}{\nu _1^2} \Bigg ( \int _0^\infty \left( P(y) y^2 - \frac{1}{2H^2(y+1)}\right) \mathrm{d}y\\&\qquad - \frac{\mathrm{ln}K}{2H(1+H)} - \frac{\mathrm{ln}2 + \mathrm{ln}H + \gamma }{2 H^2} \Bigg ) - 1. \end{aligned}$$

In the Main Text, we showed that the coefficient of variation remains bounded as K goes to zero if \(\varepsilon <{1}/{2H}\) and increases polynomially if \(\varepsilon >{1}/{2H}\). The above result implies that in the critical case \(\varepsilon ={1}/{2H}\) exhibits a slow logarithmic increase as K tends to zero.

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Bokes, P., Lin, Y.T. & Singh, A. High Cooperativity in Negative Feedback can Amplify Noisy Gene Expression. Bull Math Biol 80, 1871–1899 (2018). https://doi.org/10.1007/s11538-018-0438-y

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