Selection of mutants in a microbial population depends on multiple cellular traits. In serial-dilution evolution experiments, three key traits are the lag time when transitioning from starvation to growth, the exponential growth rate, and the yield (number of cells per unit resource). Here we investigate how these traits evolve in laboratory evolution experiments using a minimal model of population dynamics, where the only interaction between cells is competition for a single limiting resource. We find that the fixation probability of a beneficial mutation depends on a linear combination of its growth rate and lag time relative to its immediate ancestor, even under clonal interference. The relative selective pressure on growth rate and lag time is set by the dilution factor; a larger dilution factor favors the adaptation of growth rate over the adaptation of lag time. The model shows that yield, however, is under no direct selection. We also show how the adaptation speeds of growth and lag depend on experimental parameters and the underlying supply of mutations. Finally, we investigate the evolution of covariation between these traits across populations, which reveals that the population growth rate and lag time can evolve a nonzero correlation even if mutations have uncorrelated effects on the two traits. Altogether these results provide useful guidance to future experiments on microbial evolution.
The cyanobacterium Synechococcus elongatus possesses a circadian clock in the form of a group of proteins whose concentrations and phosphorylation states oscillate with daily periodicity under constant conditions. The circadian clock regulates the cell cycle such that the timing of the cell divisions is biased toward certain times during the circadian period, but the mechanism underlying this phenomenon remains unclear. Here, we propose a mechanism in which a protein limiting for division accumulates at a rate proportional to the cell volume growth and is modulated by the clock. This “modulated rate” model, in which the clock signal is integrated over time to affect division timing, differs fundamentally from the previously proposed “gating” concept, in which the clock is assumed to suppress divisions during a specific time window. We found that although both models can capture the single-cell statistics of division timing in S. elongatus, only the modulated rate model robustly places divisions away from darkness during changes in the environment. Moreover, within the framework of the modulated rate model, existing experiments on S. elongatus are consistent with the simple mechanism that division timing is regulated by the accumulation of a division limiting protein in a phase with genes whose activity peaks at dusk.
In isogenic microbial populations, phenotypic variability is generated by a combination of stochastic mechanisms, such as gene expression, and deterministic factors, such as asymmetric segregation of cell volume. Here we address the question: how does phenotypic variability of a microbial population affect its fitness? While this question has previously been studied for exponentially growing populations, the situation when the population size is kept fixed has received much less attention, despite its relevance to many natural scenarios. We show that the outcome of competition between multiple microbial species can be determined from the distribution of phenotypes in the culture using a generalization of the well-known Euler–Lotka equation, which relates the steady-state distribution of phenotypes to the population growth rate. We derive a generalization of the Euler–Lotka equation for finite cultures, which relates the distribution of phenotypes among cells in the culture to the exponential growth rate. Our analysis reveals that in order to predict fitness from phenotypes, it is important to understand how distributions of phenotypes obtained from different subsets of the genealogical history of a population are related. To this end, we derive a mapping between the various ways of sampling phenotypes in a finite population and show how to obtain the equivalent distributions from an exponentially growing culture. Finally, we use this mapping to show that species with higher growth rates in exponential growth conditions will have a competitive advantage in the finite culture.
The single-celled green algae Chlamydomonas reinhardtii with its two flagella - microtubule-based structures of equal and constant lengths - is the canonical model organism for studying size control of organelles. Experiments have identified motor-driven transport of tubulin to the flagella tips as a key component of their length control. Here we consider a class of models whose key assumption is that proteins responsible for the intraflagellar transport (IFT) of tubulin are present in limiting amounts. We show that the limiting-pool assumption is insufficient to describe the results of severing experiments, in which a flagellum is regenerated after it has been severed. Next, we consider an extension of the limiting-pool model that incorporates proteins that depolymerize microtubules. We show that this 'active disassembly' model of flagellar length control explains in quantitative detail the results of severing experiments and use it to make predictions that can be tested in experiments.
Adaptation, where a population evolves increasing fitness in a fixed environment, is typically thought of as a hill-climbing process on a fitness landscape. With a finite genome, such a process eventually leads the population to a fitness peak, at which point fitness can no longer increase through individual beneficial mutations. Instead, the ruggedness of typical landscapes due to epistasis between genes or DNA sites suggests that the accumulation of multiple mutations (via a process known as stochastic tunneling) can allow a population to continue increasing in fitness. However, it is not clear how such a phenomenon would affect long-term fitness evolution. By using a spin-glass type model for the fitness function that takes into account microscopic epistasis, we find that hopping between metastable states can mechanistically and robustly give rise to a slow, logarithmic average fitness trajectory.
MreB is an actin homolog that is essential for coordinating the cell wall synthesis required for the rod shape of many bacteria. Previously we have shown that filaments of MreB bind to the curved membranes of bacteria and translocate in directions determined by principal membrane curvatures to create and reinforce the rod shape (Hussain et al., 2018). Here, in order to understand how MreB filament dynamics affects their cellular distribution, we model how MreB filaments bind and translocate on membranes with different geometries. We find that it is both energetically favorable and robust for filaments to bind and orient along directions of largest membrane curvature. Furthermore, significant localization to different membrane regions results from processive MreB motion in various geometries. These results demonstrate that the in vivo localization of MreB observed in many different experiments, including those examining negative Gaussian curvature, can arise from translocation dynamics alone.
We study the spread of a quantum-mechanical wave packet in a noisy environment, modeled using a tight-binding Hamiltonian. Despite the coherent dynamics, the fluctuating environment may give rise to diffusive behavior. When correlations between different level-crossing events can be neglected, we use the solution of the Landau-Zener problem to find how the diffusion constant depends on the noise. We also show that when an electric field or external disordered potential is applied to the system, the diffusion constant is suppressed with no drift term arising. The results are relevant to various quantum systems, including exciton diffusion in photosynthesis and electronic transport in solid-state physics.
Membrane lysis, or rupture, is a cell death pathway in bacteria frequently caused by cell wall-targeting antibiotics. Although previous studies have clarified the biochemical mechanisms of antibiotic action, a physical understanding of the processes leading to lysis remains lacking. Here, we analyze the dynamics of membrane bulging and lysis in Escherichia coli, in which the formation of an initial, partially subtended spherical bulge (“bulging”) after cell wall digestion occurs on a characteristic timescale of 1 s and the growth of the bulge (“swelling”) occurs on a slower characteristic timescale of 100 s. We show that bulging can be energetically favorable due to the relaxation of the entropic and stretching energies of the inner membrane, cell wall, and outer membrane and that the experimentally observed timescales are consistent with model predictions. We then show that swelling is mediated by the enlargement of wall defects, after which cell lysis is consistent with both the inner and outer membranes exceeding characteristic estimates of the yield areal strains of biological membranes. These results contrast biological membrane physics and the physics of thin, rigid shells. They also have implications for cellular morphogenesis and antibiotic discovery across different species of bacteria.
Asymmetric segregation of key proteins at cell division—be it a beneficial or deleterious protein—is ubiquitous in unicellular organisms and often considered as an evolved trait to increase fitness in a stressed environment. Here, we provide a general framework to describe the evolutionary origin of this asymmetric segregation. We compute the population fitness as a function of the protein segregation asymmetry a, and show that the value of a which optimizes the population growth manifests a phase transition between symmetric and asymmetric partitioning phases. Surprisingly, the nature of phase transition is different for the case of beneficial proteins as opposed to deleterious proteins: a smooth (second order) transition from purely symmetric to asymmetric segregation is found in the former, while a sharp transition occurs in the latter. Our study elucidates the optimization problem faced by evolution in the context of protein segregation, and motivates further investigation of asymmetric protein segregation in biological systems.
Many experiments show that the numbers of mRNA and protein are proportional to the cell volume in growing cells. However, models of stochastic gene expression often assume constant transcription rate per gene and constant translation rate per mRNA, which are incompatible with these experiments. Here, we construct a minimal gene expression model to fill this gap. Assuming ribosomes and RNA polymerases are limiting in gene expression, we show that the numbers of proteins and mRNAs both grow exponentially during the cell cycle and that the concentrations of all mRNAs and proteins achieve cellular homeostasis; the competition between genes for the RNA polymerases makes the transcription rate independent of the genome number. Furthermore, by extending the model to situations in which DNA (mRNA) can be saturated by RNA polymerases (ribosomes) and becomes limiting, we predict a transition from exponential to linear growth of cell volume as the protein-to-DNA ratio increases.
For many decades, the wedding of quantitative data with mathematical modeling has been fruitful, leading to important biological insights. Here, we review some of the ongoing efforts to gain insights into problems in microbiology – and, in particular, cell-cycle progression and its regulation – through observation and quantitative analysis of the natural fluctuations in the system. We first illustrate this idea by reviewing a classic example in microbiology – the Luria–Delbrück experiment – and discussing how, in that case, useful information was obtained by looking beyond the mean outcome of the experiment, but instead paying attention to the variability between replicates of the experiment. We then switch gears to the contemporary problem of cell cycle progression and discuss in more detail how insights into cell size regulation and, when relevant, coupling between the cell cycle and the circadian clock, can be gained by studying the natural fluctuations in the system and their statistical properties. We end with a more general discussion of how (in this context) the correct level of phenomenological model should be chosen, as well as some of the pitfalls associated with this type of analysis. Throughout this review the emphasis is not on providing details of the experimental setups or technical details of the models used, but rather, in fleshing out the conceptual structure of this particular approach to the problem. For this reason, we choose to illustrate the framework on a rather broad range of problems, and on organisms from all domains of life, to emphasize the commonality of the ideas and analysis used (as well as their differences).
It is well known that the contribution of harmonic phonons to the thermal conductivity of 1D systems diverges with the harmonic chain length L (explicitly, increases with L as a power-law with a positive power). Furthermore, within various one-dimensional models containing disorder it was shown that this divergence persists, with the thermal conductivity scaling as √L under certain boundary conditions, where L is the length of the harmonic chain. Here we show that when the chain is weakly coupled to the heat reservoirs and there is strong disorder this scaling can be violated. We find a weaker power-law dependence on L, and show that for sufficiently strong disorder the thermal conductivity stops being anomalous -- despite both density-of-states and the diverging localization length scaling anomalously. Surprisingly, in this strong disorder regime two anomalously scaling quantities cancel each other to recover Fourier's law of heat transport.
We study localization of waves in a one-dimensional disordered metamaterial of bilayers composed of thin fixed length scatterers placed randomly along a homogenous medium. As an interplay between order and disorder, we identify a new regime of strong disorder where the localization length becomes independent of the amount of disorder but depends on the frequency of the wave excitation and on the properties of the fixed length scatterer. As an example of a naturally occurring nearly one-dimensional disordered bilayer, we calculate the wavelength-dependent reflection spectrum for Koi fish using the experimentally measured parameters and find that the main mechanisms for the emergence of their silver structural coloration can be explained through the phenomenon of localization of light in the regime of strong disorder discussed above. Finally, we show that, by tuning the thickness of the fixed length scatterer, the above design principles could be used to engineer disordered metamaterials which selectively allow harmonics of a fundamental frequency to be transmitted in an effect which is similar to the insertion of a half-wave cavity in a quarter-wavelength stack. However, in contrast to the Lorentzian resonant peak of a half-wave cavity, we find that our disordered layer has a Gaussian line shape whose width becomes narrower as the number of disordered layers is increased.
Most microorganisms regulate their cell size. In this article, we review some of the mathematical formulations of the problem of cell size regulation. We focus on coarse-grained stochastic models and the statistics that they generate. We review the biologically relevant insights obtained from these models. We then describe cell cycle regulation and its molecular implementations, protein number regulation, and population growth, all in relation to size regulation. Finally, we discuss several future directions for developing understanding beyond phenomenological models of cell size regulation.
In nature, microorganisms exhibit different volumes spanning six orders of magnitude 1 . Despite their capability to create different sizes, a clonal population in a given environment maintains a uniform size across individual cells. Recent studies in eukaryotic and bacterial organisms showed that this homogeneity in cell size can be accomplished by growing a constant size between two cell cycle events (that is, the adder model 2-6 ). Demonstration of the adder model led to the hypothesis that this phenomenon is a consequence of convergent evolution. Given that archaeal cells share characteristics with both bacteria and eukaryotes, we investigated whether and how archaeal cells exhibit control over cell size. To this end, we developed a soft-lithography method of growing the archaeal cells to enable quantitative time-lapse imaging and single-cell analysis, which would be useful for other microorganisms. Using this method, we demonstrated that Halobacterium salinarum, a hypersaline-adapted archaeal organism, grows exponentially at the single-cell level and maintains a narrow-size distribution by adding a constant length between cell division events. Interestingly, the archaeal cells exhibited greater variability in cell division placement and exponential growth rate across individual cells in a population relative to those observed in Escherichia coli 6-9 . Here, we present a theoretical framework that explains how these larger fluctuations in archaeal cell cycle events contribute to cell size variability and control.
Discriminating between correct and incorrect substrates is a core process in biology but how is energy apportioned between the conflicting demands of accuracy (μ), speed (σ) and total entropy production rate (P)? Previous studies have focussed on biochemical networks with simple structure or relied on simplifying kinetic assumptions. Here, we use the linear framework for timescale separation to analytically examine steady-state probabilities away from thermodynamic equilibrium for networks of arbitrary complexity. We also introduce a method of scaling parameters that is inspired by Hopfield's treatment of kinetic proofreading. Scaling allows asymptotic exploration of high-dimensional parameter spaces. We identify in this way a broad class of complex networks and scalings for which the quantity σ*ln(μ)/P remains asymptotically finite whenever accuracy improves from equilibrium, so that μ_eq/μ→0. Scalings exist, however, even for Hopfield's original network, for which σ*ln(μ)/P is asymptotically infinite, illustrating the parametric complexity. Outside the asymptotic regime, numerical calculations suggest that, under more restrictive parametric assumptions, networks satisfy the bound, σ*ln(μ/μ_eq)/P<1, and we discuss the biological implications for discrimination by ribosomes and DNA polymerase. The methods introduced here may be more broadly useful for analysing complex networks that implement other forms of cellular information processing.
MreB is essential for rod shape in many bacteria. Membrane-associated MreB filaments move around the rod circumference, helping to insert cell wall in the radial direction to reinforce rod shape. To understand how oriented MreB motion arises, we altered the shape of Bacillus subtilis. MreB motion is isotropic in round cells, and orientation is restored when rod shape is externally imposed. Stationary filaments orient within protoplasts, and purified MreB tubulates liposomes in vitro, orienting within tubes. Together, this demonstrates MreB orients along the greatest principal membrane curvature, a conclusion supported with biophysical modeling. We observed that spherical cells regenerate into rods in a local, self-reinforcing manner: rapidly propagating rods emerge from small bulges, exhibiting oriented MreB motion. We propose that the coupling of MreB filament alignment to shape-reinforcing peptidoglycan synthesis creates a locally-acting, self-organizing mechanism allowing the rapid establishment and stable maintenance of emergent rod shape.
In model bacteria, such as E. coli and B. subtilis, regulation of cell-cycle progression and cellular organization achieves consistency in cell size, replication dynamics, and chromosome positioning [ 1–3 ]. Mycobacteria elongate and divide asymmetrically, giving rise to significant variation in cell size and elongation rate among closely related cells [ 4, 5 ]. Given the physical asymmetry of mycobacteria, the models that describe coordination of cellular organization and cell-cycle progression in model bacteria are not directly translatable [ 1, 2, 6–8 ]. Here, we used time-lapse microscopy and fluorescent reporters of DNA replication and chromosome positioning to examine the coordination of growth, division, and chromosome dynamics at a single-cell level in Mycobacterium smegmatis (M. smegmatis) and Mycobacterium bovis Bacillus Calmette-Guérin (BCG). By analyzing chromosome and replisome localization, we demonstrated that chromosome positioning is asymmetric and proportional to cell size. Furthermore, we found that cellular asymmetry is maintained throughout the cell cycle and is not established at division. Using measurements and stochastic modeling of mycobacterial cell size and cell-cycle timing in both slow and fast growth conditions, we found that well-studied models of cell-size control are insufficient to explain the mycobacterial cell cycle. Instead, we showed that mycobacterial cell-cycle progression is regulated by an unprecedented mechanism involving parallel adders (i.e., constant growth increments) that start at replication initiation. Together, these adders enable mycobacterial populations to regulate cell size, growth, and heterogeneity in the face of varying environments.
Organisms across all domains of life regulate the size of their cells. However, the means by which this is done is poorly understood. We study two abstracted "molecular" models for size regulation: inhibitor dilution and initiator accumulation. We apply the models to two settings: bacteria like Escherichia coli, that grow fully before they set a division plane and divide into two equally sized cells, and cells that form a bud early in the cell division cycle, confine new growth to that bud, and divide at the connection between that bud and the mother cell, like the budding yeast Saccharomyces cerevisiae. In budding cells, delaying cell division until buds reach the same size as their mother leads to very weak size control, with average cell size and standard deviation of cell size increasing over time and saturating up to 100-fold higher than those values for cells that divide when the bud is still substantially smaller than its mother. In budding yeast, both inhibitor dilution or initiator accumulation models are consistent with the observation that the daughters of diploid cells add a constant volume before they divide. This adder behavior has also been observed in bacteria. We find that in bacteria an inhibitor dilution model produces adder correlations that are not robust to noise in the timing of DNA replication initiation or in the timing from initiation of DNA replication to cell division (the C + D period). In contrast, in bacteria an initiator accumulation model yields robust adder correlations in the regime where noise in the timing of DNA replication initiation is much greater than noise in the C + D period, as reported previously . In bacteria, division into two equally sized cells does not broaden the size distribution.
Establishing a quantitative connection between the population growth rate and the generation times of single cells is a prerequisite for understanding evolutionary dynamics of microbes. However, existing theories fail to account for the experimentally observed correlations between mother-daughter generation times that are unavoidable when cell size is controlled for, which is essentially always the case. Here, we study population-level growth in the presence of cell size control and corroborate our theory using experimental measurements of single-cell growth rates. We derive a closed formula for the population growth rate and demonstrate that it only depends on the single-cell growth rate variability, not other sources of stochasticity. Our work provides an evolutionary rationale for the narrow growth rate distributions often observed in nature: when single-cell growth rates are less variable but have a fixed mean, the population will exhibit an enhanced population growth rate as long as the correlations between the mother and daughter cells' growth rates are not too strong.