Abstract
Cells physiology adapts globally to changes in growth conditions. This includes changes in cell division rate, cell size, and gene expression. These global physiological changes are expected to affect noise in gene expression in addition to average molecule concentrations. Gene expression is inherently stochastic, and the amount of noise in protein levels depends on both gene expression rates and the cell division cycle.
Here, we model stochastic gene expression inside growing and dividing cells to study the effect of cell division rate on noise in gene expression. We use a modelling framework and parameters relevant to E. coli, for which abundant quantitative data is available.
We find that coupling of transcription rate (but not translation rate) with the division rate results in homeostasis of both protein concentration and noise across conditions. Interestingly, we find that the increased cell size at fast division rates, observed in E. coli and other unicellular organisms, prevents noise increase even for proteins with decreased average expression at faster growth.
We then investigate the functional importance of these regulations by considering gene regulatory networks that exhibit bistability and oscillations. We find that the topology of the gene regulatory network can affect its robustness with respect to changes in division rate in complex and unexpected ways. In particular, a simple model of persistence based on global physiological feedback predicts an increase in the persistence population at low division rates.
Our study reveals a potential role for cell size regulation in the global control of gene expression noise. It also highlights that understanding of circuits’ robustness across growth conditions is key for the effective design of synthetic biological systems.
Introduction
Microbial species can proliferate in a variety of environmental conditions. How genomes achieve this phenotypic flexibility is a fundamental biological question. Regulated gene expression is a key mechanism by which cells adapt physiologically to changing environments. For example, different types of metabolic enzymes are expressed to support growth on different carbon sources (Görke & Stülke, 2008). Despite this remarkable adaptability, the rate at which cells proliferate can vary strongly from one environment to another. For example, E. coli division rates range between 0.5 to 3.5 doublings per hour in response to different carbon sources (Taheri-Araghi et al, 2015).
In addition to specific gene regulation, changes in division rate are accompanied by global physiological changes (Figure 1), such as changes in cell size at division and gene expression. Global changes in gene expression with cellular growth rates are required to counteract the increase in dilution rate inherent to faster proliferation and maintain average protein concentrations. This global coordination of gene expression with the division rate could involve changes in transcription, translation and mRNA turnover. Experimental evidence suggests that in yeast and bacteria this coordination occurs primarily at the level of transcription (Keren et al, 2013; Gerosa et al, 2013; Berthoumieux et al, 2013; García-Martínez et al, 2016). Consistent with this, global translation rates in bacteria are less affected than transcription rates by the division rate, except at very slow proliferation rates (Klumpp et al, 2013; Dai et al, 2016). In B. subtilis, the translation rate (per mRNA) has even been found to decrease with the division rate, while the total mRNA concentration doubles as the division rate doubles (Borkowski et al, 2016). In yeast, mRNA turnover rates have been proposed to be globally regulated by the division rate (García-Martínez et al, 2016). Yet, it is unclear whether certain mechanisms of global gene expression regulation by the division rate are particularly advantageous over others for a fixed protein synthesis output.
The expression parameters of different genes do not necessarily follow the same dependency with the division rate. In fact, the proteome fraction of distinct functional classes has been shown to follow specific and simple trends with the division rate (Scott et al, 2010; Li et al, 2014; Hui et al, 2015). Fundamentally, for a given type of division rate modulation, proteins can be categorised in three classes (R, P, Q) depending on whether their proteome fraction respectively increases, decreases or is maintained with the division rate. Simple models of proteome allocation and cell physiology have shown that the changes in global protein fractions observed experimentally are consistent with the maximisation of the division rate (Molenaar et al, 2009; Scott et al, 2014; Goelzer & Fromion, 2017). For example, when nutrient conditions are varied, ribosomal proteins that constitute most of the R proteins are needed in larger amounts to support fast growth in rich media (Scott et al, 2010). A consequence of a large R sector is that other proteins will necessarily fall into the P class, as proteome fractions add up to one. The Q class contains so-called housekeeping proteins, whose proteome fraction is maintained across all conditions.
Because total protein concentration is approximately constant across conditions (Basan et al, 2015), the concentration of P proteins decreases at fast growth. Lower concentrations mean lower number of molecules per unit of volume. Intrinsic noise, which results from the random timing of biochemical reactions and depends on absolute molecule numbers rather than concentrations, could therefore be higher at fast growth. Intrinsic noise contributes to cell-to-cell variability in gene expression, which leads to non-genetic phenotypic variability (Shahrezaei & Swain, 2008). In addition, gene expression is affected by other stochastic and dynamic cellular processes, resulting in so-called extrinsic noise (Elowitz et al, 2002; Shahrezaei et al, 2008). An important source of extrinsic noise in gene expression stems from the processes associated with the cell cycle, including cell growth and cell division, as illustrated by several experimental and modelling studies that are discussed below. Mathematical modelling has suggested that random partitioning of biomolecules at cell division is an important source of noise in gene expression and hard to separate from intrinsic noise (Huh & Paulsson, 2011). Other modelling studies have highlighted the contribution of heterogeneity in cell cycle time on noise in gene expression (Johnston et al, 2012; Schwabe & Bruggeman, 2014; Antunes & Singh, 2014; Soltani et al, 2016). Also, cell cycle dependent expression and the timing of DNA replication also influences noise in gene expression in unexpected ways (Luo et al, 2013; Schwabe & Bruggeman, 2014; Peterson et al, 2015; Soltani et al, 2016). Several experimental studies have identified the cell cycle as a major source of noise in gene expression in bacteria and yeast (Cookson et al, 2010; Zopf et al, 2013; Keren et al, 2015; Walker et al, 2016). These studies suggest that gene expression noise is generally higher at lower division rates (Keren et al, 2015; Walker et al, 2016). The impact of cell division and random partitioning of molecules on the behaviour of simple circuits has also been studied by modelling (Gonze, 2013; Lloyd-Price et al, 2014; Bierbaum & Klumpp, 2015). It has been shown that simple genetic oscillators can sustain oscillation in the presence of cell division but the oscillations could be entrained by the cell cycle depending on the circuit topology (Gonze, 2013). Also, it is shown that random partitioning of biomolecules at division affects dynamics of simple circuits for example affecting stability of biological switches (Lloyd-Price et al, 2014).
Cell size is regulated both across the division cycles and between different growth conditions. Although this is a long-standing problem in cell biology, the mechanisms behind cell size homeostasis remain largely elusive. Interest for this question has been recently renewed, particularly in bacteria. Recent data suggests that many bacterial species follow a so-called adder principle, adding a constant cytoplasm volume in each division cycle, independently of their size at birth. Interestingly, cell size at division is positively correlated with division rates in both bacteria and yeast, cells becoming larger in richer environments (Schaechter et al, 1958; Turner et al, 2012). Although this is a universal observation, there is no satisfying universal explanation of why cells have evolved such regulation of cell size with growth conditions.
Global regulation of gene expression and cell size is likely to affect the dynamics and function of genetic and biochemical networks inside cells (Shahrezaei & Marguerat, 2015). A pioneering study quantified how division rate dependent global regulation of gene expression affects the average concentration of a constitutively expressed gene product, and how this in turn can affect the behaviour of simple synthetic genetic networks (Klumpp et al, 2009). Another theoretical study showed that the division rate dependence of gene expression could impact the qualitative behaviour of a synthetic oscillator circuit, the ‘repressilator’ (Osella & Lagomarsino, 2013). Moreover, the division rate regulation of a gene impacting fitness can result in non-trivial global feedback in gene regulation (Klumpp et al, 2009; Kiviet et al, 2014; Tan et al, 2009). However, theoretical insights on how global regulation of gene expression and cell size with growth conditions impacts noise in gene expression and therefore the behaviour of biochemical circuits are still largely lacking.
In this study, we shed light on the regulation of noise in gene expression across growth conditions by integrating existing data in the bacterium E. coli on global regulation of gene expression and cell size into detailed computational models of stochastic gene expression in growing and dividing cells. We then use examples of some simple genetic networks to illustrate how the changes in gene expression noise across growth conditions affects the dynamics of cellular systems.
Results
Stochastic gene expression in growing and dividing cells
To fully capture the effect of cell cycle on noise in gene expression, we model the stochastic expression of a single gene in growing and dividing cells (Figure 2 A-B, Supplemental Figure 1-A). Transcription, mRNA degradation and translation are represented by single stochastic reactions. Corresponding rates are noted km, γm and kp respectively. Because the majority of E. coli proteins are stable, we first neglect protein degradation. During the cell cycle, we assume cell size increases exponentially at a fixed rate, that results in a decrease of the concentration of the mRNA and the protein when their numbers do not change. We model cell division as a discrete event that splits the cell volume in two, and each molecule is randomly partitioned between daughter cells with a probability matching their inherited volume fraction. In our simulations, we keep only one of the two daughter cells, therefore reproducing the popular mother machine experimental setting (Wang et al, 2010).
Cellular growth rate, cell size at division, and cell size at birth are all known to vary between individual cells even in identical, tightly controlled conditions. Variability in size at birth arises from variability in the mother cell size at division but also from imperfect volume splitting between the two daughter cells. To realistically account for this variability, we use the noisy linear map (NLM) model (see Methods and Supplemental Figure 1), a recent phenomenological model of cell size control that captures the variability in cell size at birth and division observed experimentally as well as their correlation within individual cell cycles (Tanouchi et al, 2015; Jun & Taheri-Araghi, 2015). The degree of this correlation is related to the mechanims underlying cell size homeostasis. For example, a noisy linear map with the parameter a equal to 1 corresponds to an adder strategy, where a fixed cytoplasm volume is added to the cell between each division. Alternatively, a parameter a equal to zero corresponds to a sizer strategy, where cell division is triggered at a fixed size (Jun & Taheri-Araghi, 2015).
A priori, it is possible that the NLM parameters that best describe a given single-cell dataset could change with growth conditions. Therefore, we have inferred the parameters of the NLM from a recent mother machine dataset of cells grown in 7 different carbon sources supporting a wide range of division rates (Taheri-Araghi et al, 2015). We find that NLM parameters can indeed change with the division rate (Supplemental Figure 1). As expected, b strongly increases with the division rate (the average size at division is given by ). Notably, the slope parameter is significantly lower than 1 at slow growth, consistently with another study reporting a deviation towards a sizer strategy (a < 1) in slow regimes (Wallden et al, 2016). In addition, individual cell growth rates are well described by normal distributions in all conditions. Based on that analysis, we derive linear functions describing all NLM parameters as a function of the division rate (Supplemental Figure 1). This enables us to realistically model growth and division at the single cell level over a wide range of division rates and investigate their effects on gene expression noise.
Before starting to explore effect of division rate on noise in gene expression and using the NLM parameters extracted from the data, we first explore the effect of these parameters on the gene expression noise for a fixed growth condition, as this has not been explored before. In Figure 2-C, we show protein number and concentration noise (CV) at cell birth (immediately after cell division and at the beginning of the cell cycle) as the noise in final size (σ1), noise in size partitioning (σ2) and a are varied. Large noise in NLM noise (σ1 or σ2) results in an increased noise in protein number noise at the beginning of the cell cycle (Figure 2-C). This is due to partitioning noise as this increased protein number noise is mostly decayed in the middle of cell cycle (Supplemental Figure 2). Also, protein concentration noise is not so much affected by NLM noise as we assume probability of random partitioning of biomolecules is proportional to the inherited volume of the daughter cells after division. For values of a greater than one size control is not very effective in filtering noise in cell size and there is an increased size variability for large a and large NLM noise (σ1 or σ2) (Modi et al, 2017). As a result the protein concentration noise that directly depends on cell volume shows an increase at large a and large NLM noise. Overall, these results show that the physiological range of NLM parameters across growth conditions (Supplemental Figure 1) are not expected to produce strong effects in noise gene expression.
In the results shown in Figure 2-C, we have assumed the reaction propensities for transcription, translation and mRNA decay are independent of cell volume. In Supplemental Figure 3, we show the impact of a cell size-dependent transcription rate. Interestingly, in this case, the protein concentration noise is reduced and becomes independent of the NLM parameters. We obtain very similar results if we assume translation rate is size-dependent (not shown). Size dependence of transcription rate has been recently reported in eukaryotes (Padovan-Merhar et al, 2015; Kempe et al, 2015), while similar evidence in prokaryotes is lacking. Therefore, in this work we assume cell size independent propensities for all first-order reactions (but volume dependency for bi-molecular reaction propensities is accounted for). Also, we focus on protein concentration noise (physiologically more relevant than molecule numbers) and across newly born cells (to eliminate cell cycle stage contributions, similar trends are seen in the middle of the cell cycle).
Expression noise depends on division rate even when protein concentration is maintained
We consider first genes whose protein concentration stays constant when the division rate changes (i.e. proteins belonging to the Q class). Interestingly, this requires that at least one of the gene expression rates km (transcription rate), γm (mRNA degradation rate) or kp (translation rate per mRNA) changes with the division rate to compensate for increased dilution of mRNA and protein molecules.
Using our model and typical values for gene expression rates at 2 doublings per hour as a baseline, we computed the change in protein concentration noise with the division rate when average concentration is maintained either by adapting the transcription rate only (Figure 3-A) or the translation rate per mRNA only (Figure 3-B). To investigate the contribution of distinct sources of noise and of variability in cell size to protein concentration noise we consider multiple scenarios in which different sources of variability are turned off (colour codes in Figures 3-A and 3-B).
Our simulation results reveal that maintaining average protein concentration by adjusting transcription or translation to the division rate leads to very different behaviours of the protein concentration noise. We find that the empirically observed increase of cell size with division rate strongly contributes to these behaviours. In the case of transcription rate adjustment, protein noise sharply decreases with the division rate. A milder decrease is also observed when cell size is kept constant across division rates. In the case of translation rate adjustment, protein noise increases with the division rate instead, whether cell size changes or not.
To better understand these results, we looked at how mRNA numbers change with the division rate in the different situations (bottom left plots in Figures 3-A and 3-B). When transcription adjusts to the division rate in order to maintain average protein expression, mRNA numbers increases with the division rate. As mRNA noise (mRNA numbers are typically much lower than protein numbers) is a major contributor of protein noise, an increase in mRNA numbers results in a decrease in protein noise. However when instead translation adjusts to the division rate, mRNA numbers remain mostly unchanged. This is possible, because mRNA degradation rates are large compared to the division rate, resulting in mRNA numbers being less sensitive to dilution than protein numbers. Despite little change in mRNA numbers and hence mRNA noise, the increase in protein noise can be explained by a higher propagation of the mRNA noise to protein, since contribution of transcription to protein noise depends on the ratio of mRNA lifetime (which is mostly constant) and protein lifetime (which is set by the dilution rate, itself set by the division rate) (Swain et al, 2002).
While the relative contribution of distinct noise sources (stochastic gene expression, partitioning noise, variability in cell growth rate, cell division size and cell birth size) to total protein noise can change with the division rate, we find that the contribution of stochastic gene expression is predominant at all division rates (Supplemental Figure 4). For the case of transcription adjusting to division rate, we find the contribution of partitioning noise is relatively constant across division rates, while contribution of LNM noise increases several folds at fast division rates. In contrast for the case of transcription adjusting to division rate, we find the contribution of partitioning noise significantly decreases at fast division rates, while contribution of LNM noise remains relatively constant.
In summary, our simulations demonstrate that for genes with typical expression parameters at intermediate division rates, maintaining a constant protein concentration across growth conditions by adjusting transcription to the division rate leads to a decrease of protein noise. In contrast, adjusting translation to the division rate increases protein noise levels.
Increase of cell size with the division rate prevents noise increase for constitutively expressed proteins despite a decrease in average concentration
The results described above concern proteins belonging to the Q category, whose average concentration is maintained constant independently of the division rate. Klumpp and colleagues have shown that constitutively expressed proteins instead belong to the P category: their concentration is decreased at fast growth (Klumpp et al, 2009). The transcription rate of constitutively expressed genes strongly increases with the division rate, while mRNA degradation rate and translation rate per mRNA remain relatively constant (Klumpp et al, 2009). However, this is not sufficient to balance both increased dilution and increased cell size (Klumpp et al (2009), Supplemental Figure 5 and Figure 4 top left plot).
Remarkably, using parameters of gene expression from (Klumpp et al, 2009) (see Methods and Supplemental Figure 5), we find that protein noise decreases with division rate, despite the strong decrease in average protein concentration (Figure 4). Cell size increase with division rate is a key contributor to this behaviour. Assuming that increased expression noise for P proteins at fast growth is deleterious, this observation could explain why increased cell size at fast division rates is a universally conserved feature of unicellular organisms.
In the case of a P protein, similarly to the case of Q protein above, we find that the relative contribution contribution of stochastic gene expression is predominant at all division rates (Supplemental Figure 4). However, contribution of both partitioning noise and size and growth rate variability increases moderately at fast division rates.
Impact of division rate on the behaviour of an oscillator circuit
Changes in average expression and noise of individual proteins with the division rate in response to environmental changes is likely to impact the behaviour of genetic circuits (Klumpp et al, 2009). Even when the protein average expression (in isolation, i.e. without the circuit-specific regulations) is maintained, the expression noise can still change (Figure 2) meaning that circuit behaviour could depend on the division rate (Shahrezaei & Marguerat, 2015).
To investigate these effects, we first consider a two proteins oscillator circuit recapitulating essential features of circadian clocks (Figure 5-A) (Vilar et al, 2002). An actively degraded activator protein (A) promotes its own transcription as well as the transcription of a stable repressor protein (R) by promoter binding. R can also binds A, preventing it to bind promoters. This circuit can lead to oscillations as illustrated in Figure 5-B. A detailed analysis of why oscillations arise is beyond the scope of this study and has been explored before (Guantes & Poyatos, 2006; Kut et al, 2009). Briefly, because R competes with promoters for the binding of A, when the amount of free R is large only basal transcriptional activity for both genes is possible. Because R is stable, such a state can last until dilution and partitioning renders free R levels too low to efficiently prevent promoter binding by A. Promoter activation leads to a burst of A by auto-activation, but R levels eventually rise because A also promotes R transcription. When R levels are sufficient to efficiently compete with A promoter binding, a novel cycle starts.
We asked how the circuit behaviour was affected when division rate modified. We first assume that basal transcription, translation and mRNA degradation follows the same dependency as constitutively expressed proteins (i.e. P proteins, as in Figure 4), and that the fold-change increase of transcription rate when the promoter is activated by A is independent of the division rate. The resulting changes in circuit behaviour with the division rate are shown in Figure 5-C (black lines). The average period increases as the division rate decreases because dilution is an important driver of the oscillations. The average amplitude of free R oscillations is also strongly dependent on the division rate, and decreases as the division rate increases. This is consistent with P expression, although different behaviours are in theory possible because of gene regulation. The noise in circuit behaviour changes as well with the division rate. Specifically, noise in period and amplitude of the oscillations display ‘U’ shape dependencies with the division rate, with lower noise close to the reference division rate of 2 doublings per hour. In summary, constitutive expression (typical of P proteins) leads to changes in average behaviour and a strong increase in noise of an oscillatory circuit at very low or very high division rates.
We then investigated whether Q expression of the circuit components could increase the robustness of oscillations in response to changes in division rate. As in Figure 3 we consider two modes of Q expression, either by transcriptional adjustment (blue) or translational adjustment (red). Both modes could maintain the average amplitude of oscillations in a narrow range, but the average period remained strongly dependent on the division rate (Figure 5-C). While both modes resulted in identical changes in circuit average behaviour, they led to slightly different dependencies of noise in oscillations with the division rate. The division rate with the minimal noise in amplitude is around 2.3 doublings per hour for transcriptional adjustment and around 1.5 doublings per hour for translational adjustment. In summary, Q expression increased robustness of oscillations compared to constitutive (P) expression, but it is not sufficient to make the oscillator’s period independent of the division rate. Q expression via transcriptional or translational adjustment led to similar, but not identical changes of noise in oscillations with the division rate.
Impact of division rate on the behaviour of the toggle switch
We investigate next a simple synthetic circuit known to exhibit bistability: the toggle switch (Gardner et al, 2000), in which two proteins repress each other’s transcription (Figure 6-A). We asked first whether different circuit behaviours, namely the existence of bistability, the occupancy of the states, and the switching rates between states, were affected by changes in division rate and adjustment of transcription or translation to division rates. To this end, we consider simple model assumptions that are sufficient to generate stochastic switching between different states (Methods) with typical parameter values.
We found that the circuit could exhibit bistability (Figure 6-B,C) over the considered range of division rates for constitutive (P) expression as well as for Q expression by transcriptional or translational adjustment. However, in all cases the circuit behaviour strongly depends on the division rate (Figure 6-C), as illustrated by the change in ON state occupancy (the circuit is ON when one of the two proteins, the reporter, is in the high expression state). Interestingly, the change of behaviour is very different for different modes of Q expression: for translational adjustment, the ON state occupancy decreases with the division rate (in a fashion very similar to P expression). However, an opposite behaviour is observed for Q expression via transcriptional adjustment as ON state occupancy becomes positively correlated with division rate..
The ON state occupancy reflects the balance between stochastic switching in and out of this state. These rates are both dependent on the division rate (Figure 6-C, middle and right plots). We find that the switching rates increase with the division rate that could suggest random partitioning of mRNA and protein molecules, which is more frequent at high division rates, favours switching as also reported in another study (Lloyd-Price et al, 2014). In addition, the observation that at fast growth the OFF → ON rate rises the most sharply for Q expression via translational adjustment is consistent with the high level of protein noise for this mode of regulation at fast division rates (Figure 3-B).
When gene expression feedbacks on growth: the case of toxin-mediated growth inhibition
So far, the circuits we have considered respond to changes in division rate but they don’t impact cell physiology and growth. However, many natural circuits and some synthetic circuits do influence cell physiology, for example by regulating cell metabolism or cell cycle progression. Even when synthetic circuits are not designed to impact cell physiology, they often do by competing with core cellular processes for global cellular resources, and this has become a major concern for synthetic biologists (Ceroni et al, 2015).
In prokaryotes, well-known examples of gene expression feeding back on growth are toxin-antitoxin systems. These systems are involved in bacterial persistence, where a very small subpopulation of slow growing cells naturally arises among a normally growing population. A minimal model, where a single protein is toxic for growth was found to be sufficient to generate growth bistability (Klumpp et al (2009), Tan et al (2009), Rocco et al (2013), and Figure 7-B). Here we investigate the behaviour of this kind of model (Figure 7) when both the maximal growth rate reached by a toxin-free cell and the dependency of the transcription rate with the cell growth rate are varied.
For each parameter set enabling growth bistability (coloured pixels in Figure 7-C), we computed the occupancy of the fast state (Figure 7-C, left) and the switching rates between the slow and fast states (Figure 7-C, middle and right). The occupancy of the fast growing state decreases when the maximal growth rate decreases (Figure 7-C, moving from right to the left), and this behaviour is independent of the dependency of the toxin transcription rate to the cell division rate (i.e. the value of kmslope). Therefore, the system will naturally respond to less favourable growth conditions by increasing the time spent in the slow state.
Discussion
In this study, we have used detailed simulations of stochastic gene expression in growing and dividing bacteria to investigate the role of division rate in protein noise and dynamics of genetic networks. Our simulations are constrained by data available for E. Coli related to division rate regulation of constitutive gene expression (Klumpp et al, 2009) and single-cell data related to cell size control (Taheri-Araghi et al, 2015). For a constitutively expressed gene, we find that coupling transcription but not translation to division rate results in lower protein noise levels. Interestingly, existing data seem to suggest that global regulation of gene expression with division rate mostly acts at the level of transcription (Keren et al, 2013; Gerosa et al, 2013; Berthoumieux et al, 2013; García-Martínez et al, 2016), consistent with the idea that lower noise levels are beneficial, or even necessary, at fast growth. However, regulation at the level of translation has also been observed (Borkowski et al, 2016), which, coupled to transcriptional regulation, could result in non-trivial interplay in terms of gene expression noise regulation.
An important factor that helps to minimise noise in gene expression at fast division rate is increased cell size. Large cell sizes in growth conditions with fast division rate results in higher overall number of mRNA and protein molecules, and reduce noise in gene expression. This is particularly relevant for the regulation of noise in gene expression for proteins belonging to P category (Figure 1) as their concentration go down at high division rates. Based on these results, we propose a possible evolutionary reason for microbial cells (bacteria and yeast) to grow bigger at fast growth is to reduce gene expression noise, which is presumably more detrimental to fitness at fast growth (Shahrezaei & Marguerat, 2015). At the mechanistic level, the division rate regulation of cell size could be implemented via the division rate regulation of gene expression for proteins involved in cell size control (Basan et al, 2015; Bertaux et al, 2016).
Our simulations included physiologically relevant levels of partitioning noise, size variability and growth variability. Overall, we observe that the contribution of these factors to protein noise is small but that it tends to vary with the division rate for the different cases considered. We also observed the noise in molecular numbers and concentrations do not always behave similarly, as the later directly depends on cell volume. Interestingly, we find that if transcription rate scales with cell size as recently reported in eukaryotes (Padovan-Merhar et al, 2015; Kempe et al, 2015), the concentration noise becomes independent of noise in cell size control mechanism. In bacteria, there has not been a careful investigation of transcription scaling with cell size and in the absence of such reports we have assumed cell size independent reaction propensities thoughout this study. We also did not model the contribution of DNA replication to protein concentration noise, but its impact has been found experimentally to be very small (Walker et al, 2016).
We then tested how dynamics of simple biochemical networks respond to division rate. As shown by the seminal work of Klumpp et al (2009), we find overall that division rate regulation of concentration of P proteins can change the average behaviour of biochemical networks significantly. But, as discussed below, we find that even when proteins in the network have a Q regulation, the changes in noise properties of the individual gene expression can significantly alter the mean and noise properties of the system.
In the case of a genetic oscillator, we find changes in gene expression and cell size with the division rate can impact the behaviour of oscillatory circuits in a non-trivial manner. Namely, large changes of average expression with the division rate for constitutive expression (P) of circuit components render circuit behaviour sensitive to the division rate. However, maintaining constant expression of circuit components (for example via transcriptional or translational adjustment) does not guarantee full robustness of circuit behaviour against changes in division rate. Robustness might require more complex, circuit-specific dependencies of gene expression with the division rate, or even specific circuit architecture (Paijmans et al, 2016). Interestingly, we observed a ‘U’ shape dependency of noise on division rate suggesting that there could be an optimally robust growth condition for a specific network design and parameter combination, which is relevant to appropriate function of natural biochemical systems or synthetic systems.
The toggle switch circuit behavior is strongly dependent on the division rate and on the type of gene expression dependency with the division rate. So, this suggests the simple toggle switch circuit is not going to perform robustly across growth conditions. As for the oscillator circuit, maintaining average expression is not sufficient to generate a division rate independent behaviour. Moreover, this example shows that even when average expression is maintained, whether it is maintained via adjustment of transcription or translation matters, as the circuit behaves differently in either situation.
In the case of simple models of persistence induced by the expression of a toxic protein in single growing and dividing cells, we could investigate the impact of growth conditions and gene expression dependency with the cell growth rate on the emergence of growth bistability. The role of growth conditions in prevalence of persister cells is a very relevant problem as the growth conditions of bacteria during infection are likely to be altered by the immune system and therapeutic treatments for instance. So, to validate our simple modelling results, it would be interesting to assess quantitatively, if and how growth conditions regulate the probability of the non-growing persistence phenotype.
In molecular systems biology, we use models of biochemical networks to validate our mechanistic understanding of the system under study. We propose that such models should be tested also against data collected across cellular division rates. If the behaviour of the system is observed to be robust to growth conditions, then our models should be able to capture this robustness. Conversely, describing the ways in which the system behaviour changes across growth conditions is key to refine our models and therefore our mechanistic understanding of the system under study.
In synthetic biology, we often desire to build a system that either functions robustly at a particular growth condition or across a range of growth conditions. Our study shows that stochastic models of synthetic biochemical networks in growing and dividing cells coupled with data on the regulation of gene expression across division rates are essential to optimal design of system topologies that achieve robustness against changes in cellular division rates.
Methods
Modelling
We describe here our basic model for gene expression in growing and dividing cells. mRNA molecules are randomly synthetized and degraded at rate km and γm respectively. Stochastic synthesis of protein from each mRNA occurs at rate kp. Protein molecules are assumed to be stable (except for A in the oscillator circuit). Cell volume is growing exponentially at a fixed rate between Vbirth and Vdiv = 2 Vbirth, then cell division is triggered (for the case including cell size control and variability see below). At cell division, molecules are randomly split between daughter cells and the volume is halved. In simulations, only one of the daughter cell is considered for further simulation (hence mimicking the ‘mother machine’ microfluidic experiments for a symmetrically dividing cell (Wang et al, 2010). Throughout, noise is quantified by using coefficient of variability (CV), which is defined as standard deviation divided by the mean.
Reference gene expression parameters
Realistic (Taniguchi et al, 2010) parameters for E. coli gene expression have been used (km = 0.28 min−1, γm = 0.14 min−1, kp = 0.94 min−1, µ = 2 doublings/hr). This corresponds to a mRNA half-life of 5 min, an average mRNA number at birth of 1 molecule and an average protein number at birth of 50 molecules.
Realistic modelling of cellular growth rate and cell size variability across growth conditions with noisy linear maps
We use noisy linear maps (Tanouchi et al, 2015) with parameters inferred from mother machine data in different growth conditions (Taheri-Araghi et al, 2015). See Supplemental Figure 1 for model description. a and b are estimated by linear regression of Vdiv vs Vbirth (the data contains around 100K cell cycles per condition). σ1 is by definition related to the residual of this regression. σ2 is estimated from the variance of where is the birth size recorded just after the division at Vdiv.
Modelling Q expression by transcriptional or translational adjustment
For a stable protein, it is possible to derive an analytical expression for the average number of protein molecules at birth:
This expression was used to compute the transcription or translation rate achieving a given average protein concentration (Figure 3 and 4). In the case of active protein degradation (as for A for the oscillator circuit studied in Figure 5), we used simulations and the MATLAB scalar optimization function fminsearch to compute the transcriptional or translational rate adjustment enabling to maintain a constant average concentration at birth.
Modelling P expression
For Figure 4, we have used division rate dependencies of gene expression parameters from (Klumpp et al, 2009) as illustrated in (Supplemental Figure 5). The dependencies were used as a relative scaling with respect to the reference gene expression parameters at 2 doublings per hour. For modelling P expression in the oscillator circuit (Figure 5), for simplicity we simply used the effective transcription rate division rate dependency (the cell size dependency being given by the noisy linear maps) as change in translation rate per mRNA or mRNA degradation rate are small.
Oscillator circuit
The model structure and parameterization is adapted from (Vilar et al, 2002). The A protein can transcriptionally activate its own expression as well as the expression of another protein R by promoter binding. A is short-lived while R is stable. A and R can form a complex. The same model reactions were used, but we also explicitly model growth and division (including random partitioning of free A and free R, but we do not model gene replication and consider a single copy of each promoter which is always inherited by daughter cells). The volume dependency of bi-molecular reactions is also accounted for. As reference parameters (i.e. corresponding to an intermediate E. coli division rate of 2 doublings per hour, at which optimal circuit behavior should be obtained), we used the same parameters as Vilar and colleagues, except that the R degradation rate was set to 0 (the original value, corresponding to a ~200 min half-life, was accounting for dilution only), the active degradation rate of A was scaled up to maintain a constant ratio with the division rate, the R translation rate was scaled up by the same factor, and all transcription rates were scaled by this factor (~7).
The resulting values are:
To compute the period and amplitude of oscillations in free R concentration, we used the MATLAB function findpeaks on very long (200K minutes) mother machine traces, requiring a minimum peak amplitude of 25% of the maximum value in the trace. We verified visually the behavior of the peak detection algorithm for each simulation.
Toggle switch circuit
The model structure and parameters are completely symmetric for the two proteins repressing each other. There is no cooperativity in the repression, as it is not required to obtain stochastic switching, consistently with (Lipshtat et al, 2006). As for the oscillator circuit, the volume dependency of bi-molecular reactions (only promoter binding here) was accounted for. We assumed that transcription is completely blocked when the promoters are bound, and that the promoter binding and unbinding rates are independent of the division rate.
The reference parameter values are:
Detection of bistability (always the case for simulations shown in Figure 5), threshold identification and computation of switching rates were performed as follows. A very long (500 thousands hours of biological time) single-lineage trace (one output every 15 minutes) of the free A concentration is obtained by simulation. This trace is then discretized into 50 equal size bins from zero to the maximal value of the trace. The following algorithm is then applied on this discretized distribution: (1) identify the highest mode (i.e. the most populated bin); (2) iteratively identify next highest mode and ask whether they are corresponding to a neighbor bin of the highest mode (then it is not the second mode of a bimodal distribution) OR if there exists populated, lower height bins in-between (indicative of bimodality); (3) in the latter case, to avoid incorrect detection of bimodality because of finite sampling of the distribution, the secondary mode is required to be more than 5% of what an uniform distribution would give.
Growth bistability caused by expression of a toxic protein
As previously, stochastic gene expression of a protein is simulated in growing and dividing cells. However, the protein is a toxin inhibiting cell growth: the instantaneous growth rate of the cell µcell is a decreasing Hill function of the toxin concentration (hence it is not anymore constant during the cell cycle). Also, the impact of growth conditions is not modeled anymore with condition-specific noisy linear maps, as they are not adapted to situations with very heterogeneous growth rates between cells in a given condition. We rather use a parameter µmax representing the toxin-free cellular growth rate. For simplicity, to model cell division size and its variability we use a single noisy linear map across growth conditions. Finally, to represent the dependency of gene expression with the cell growth rate, we assume that the toxin transcription rate is a linear function of µcell. The reference parameter values are:
Note that because kmslope = 0, the positive feedback toxin → growth slow down → more toxin is only mediated by a change of dilution (as in (Rocco et al, 2013)). Also note that it is necessary to assume that protein degradation is non-zero to allow bistability, as otherwise exit of the slow state is impossible.
For Figure 7-C, for each parameter set, the existence of bistability, threshold identification and switching rates computation for the instantaneous cell growth rate µcell were performed as for the toggle switch circuit analysis (except that simulation duration for each single-lineage trace was 60 thousands hours of biological time, with one output every 10 minutes, and the number of bins used was 20).
Grey indicates parameter sets for which the lineage simulation of 60 thousands hours (~120 thousands generations) either did not lead to a bimodal distribution of µcell, or did lead to such bimodal distribution, but with less than 10 switches fast → slow → fast, preventing an accurate estimate of switching rates in reasonable computational time.
Simulation algorithm
We describe here the general simulation algorithm used for all models. Between fixed timesteps (6 seconds), cell volume is considered constant, and the Gillespie algorithm is used to simulate stochastic molecular reactions (more sophisticated simulation methods exist (Lu et al, 2004; Shahrezaei et al, 2008), but this one is simple to implement and accurate as long as the timestep is small enough). Then, the cell volume is updated according to the instantaneous exponential growth rate, it is checked whether cell division should occur, and if so, cell division and molecules partitioning is realized. The code used for all simulations is available on GitHub: https://github.com/ImperialCollegeLondon/coli-noise-and-growth.
Acknowledgments
We would like to thank Philipp Thomas and Marc Sturrock for feedback on our manuscript and members of the Marguerat and Shahrezaei groups for discussions. We thank Suckjoon Jun for sharing the E. coli size data across growth conditions. We acknowledge Matthew Robb for his preliminary work on this project during his PhD. This work is supported by a Leverhulme Research Project Grant (RPG-2014-408) awarded to SM and VS. SM is supported by the UK Medical Research Council.