Impact of division rate and cell size on gene expression noise

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.


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Microbial species can proliferate in a variety of environmental conditions. How genomes 35 achieve this phenotypic flexibility is a fundamental biological question. Regulated gene 36 expression is a key mechanism by which cells adapt physiologically to changing environ-37 ments. For example, different types of metabolic enzymes are expressed to support growth 38 on different carbon sources (Görke & Stülke, 2008). Despite this remarkable adaptability, 39 the rate at which cells proliferate can vary strongly from one environment to another. For  Cellular growth rate, cell size at division, and cell size at birth are all known to vary 145 between individual cells even in identical, tightly controlled conditions. Variability in size 146 at birth arises from variability in the mother cell size at division but also from imperfect 147 volume splitting between the two daughter cells. To realistically account for this variability, Q" R" P" total proteome Figure 1: Global cellular factors affecting gene expression noise that depend on growth conditions. Nutrient quality can increase the population doubling rate by promoting growth and division of individual cells. This leads to increased dilution of molecules, and more frequent random partitioning of molecules between daughter cells. Because faster growth requires a higher rate of cell mass production, rates of mRNA and protein expression increase globally with the division rate. However, the relative changes in mRNA and protein expression rates is gene-dependent because the proteome composition is reshaped when the division rate changes (Scott et al, 2014). For example, the fraction of ribosomal proteins (R proteins) will increase with the division rate while the fraction of metabolic enzymes (and other P proteins) will decrease, the fraction of house keeping proteins (and other Q proteins) remain constant (Scott et al, 2010). Cell size as well is known to increase with the division rate in response to nutrient-based modulations (Schaechter et al, 1958;Basan et al, 2015). All those factors affect both average expression and expression noise in a non-trivial manner.

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A priori, it is possible that the NLM parameters that best describe a given single-cell 158 dataset could change with growth conditions. Therefore, we have inferred the parameters 159 of the NLM from a recent mother machine dataset of cells grown in 7 different carbon 160 sources supporting a wide range of division rates (Taheri-Araghi et al, 2015). We find 161 that NLM parameters can indeed change with the division rate (Supplemental Figure   162 1). As expected, b strongly increases with the division rate (the average size at division 163 is given by 2b 2−a ). Notably, the slope parameter a is significantly lower than 1 at slow 164 growth, consistently with another study reporting a deviation towards a sizer strategy 165 (a < 1) in slow regimes (Wallden et al, 2016). In addition, individual cell growth rates   Other parameters are kept constant at reference values, except b that changes with a such that the average size at birth is constant. Black crosses indicate empirical ranges estimated from mother machine data (see Methods and Supplemental Figure 1).

Expression noise depends on division rate even when protein concentration is
202 maintained 203 We consider first genes whose protein concentration stays constant when the division rate  tutively expressed proteins despite a decrease in average concentration 252 The results described above concern proteins belonging to the Q category, whose average

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In the case of a P protein, similarly to the case of Q protein above, we find that the relative 2) meaning that circuit behaviour could depend on the division rate (Shahrezaei & 278 Marguerat, 2015).

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To investigate these effects, we first consider a two proteins oscillator circuit recapitulating as the division rate decreases because dilution is an important driver of the oscillations.

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The average amplitude of free R oscillations is also strongly dependent on the division 300 rate, and decreases as the division rate increases. This is consistent with P expression,   321 We investigate next a simple synthetic circuit known to exhibit bistability: the toggle values.

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We found that the circuit could exhibit bistability ( Figure 6-B,C) over the considered Change of the toggle-switch behaviour, quantified by the average time spent in the ON state and the switching rates between the two states, as a function of division rate. The black curve corresponds to P expression as in Figure 3, the blue and red curves corresponds to constant average expression maintained either transcriptionally or translationally, as in Figure 2-C,D. Note that when the concentration of one protein type is low, the other is not necessarily high. This is why the ON state occupancy is not always 50% despite the symmetry between the two proteins.  cell size independent reaction propensities thoughout this study. We also did not model 405 the contribution of DNA replication to protein concentration noise, but its impact has 406 been found experimentally to be very small (Walker et al, 2016). 407 We then tested how dynamics of simple biochemical networks respond to division rate. conditions regulate the probability of the non-growing persistence phenotype.

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In molecular systems biology, we use models of biochemical networks to validate our 441 mechanistic understanding of the system under study. We propose that such models 442 should be tested also against data collected across cellular division rates. If the behaviour 443 of the system is observed to be robust to growth conditions, then our models should be 444 able to capture this robustness. Conversely, describing the ways in which the system 445 behaviour changes across growth conditions is key to refine our models and therefore our 446 mechanistic understanding of the system under study.

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In synthetic biology, we often desire to build a system that either functions robustly at 448 a particular growth condition or across a range of growth conditions. Our study shows 449 that stochastic models of synthetic biochemical networks in growing and dividing cells 450 coupled with data on the regulation of gene expression across division rates are essential 451 to optimal design of system topologies that achieve robustness against changes in cellular 452 division rates.   Oscillator circuit 503 The model structure and parameterization is adapted from (Vilar et al, 2002 To compute the period and amplitude of oscillations in free R concentration, we used 519 the MATLAB function findpeaks on very long (200K minutes) mother machine traces, 520 requiring a minimum peak amplitude of 25% of the maximum value in the trace. We 521 verified visually the behavior of the peak detection algorithm for each simulation.

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The model structure and parameters are completely symmetric for the two proteins 524 repressing each other. There is no cooperativity in the repression, as it is not required to 525 obtain stochastic switching, consistently with (Lipshtat et al, 2006). As for the oscillator 526 circuit, the volume dependency of bi-molecular reactions (only promoter binding here) was 527 accounted for. We assumed that transcription is completely blocked when the promoters 528 are bound, and that the promoter binding and unbinding rates are independent of the 529 division rate.

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The reference parameter values are:  Grey indicates parameter sets for which the lineage simulation of 60 thousands hours 565 (~120 thousands generations) either did not lead to a bimodal distribution of µ cell , or did 566 lead to such bimodal distribution, but with less than 10 switches fast → slow → fast, 567 preventing an accurate estimate of switching rates in reasonable computational time.

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Simulation algorithm 569 We describe here the general simulation algorithm used for all models. Between fixed