Publications

A critical cell cycle checkpoint for most bacteria is the onset of constriction when the septal peptidoglycan synthesis starts. According to the current understanding, the arrival of FtsN to midcell triggers this checkpoint in Escherichia coli. Recent structural and in vitro data suggests that recruitment of FtsN to the Z-ring leads to a conformational switch in actin-like FtsA, which links FtsZ protofilaments to the cell membrane and acts as a hub for the late divisome proteins. Here, we investigate this putative pathway using in vivo measurements and stochastic cell cycle modeling at moderately fast growth rates. Quantitatively upregulating protein concentrations and determining the resulting division timings shows that FtsN and FtsA numbers are not rate-limiting for the division in E. coli. However, at higher overexpression levels, they affect divisions: FtsN by accelerating and FtsA by inhibiting them. At the same time, we find that the FtsZ numbers in the cell are one of the rate-limiting factors for cell divisions in E. coli. Altogether, these findings suggest that instead of FtsN, accumulation of FtsZ in the Z-ring is one of the main drivers of the onset of constriction in E. coli at faster growth rates.

Mechanical energy, specifically in the form of ultrasound, can induce pressure variations and temperature fluctuations when applied to an aqueous media. These conditions can both positively and negatively affect protein complexes, consequently altering their stability, folding patterns, and self-assembling behavior. Despite much scientific progress, our current understanding of the effects of ultrasound on the self-assembly of amyloidogenic proteins remains limited. In the present study, we demonstrate that when the amplitude of the delivered ultrasonic energy is sufficiently low, it can induce refolding of specific motifs in protein monomers, which is sufficient for primary nucleation; this has been revealed by MD. These ultrasound-induced structural changes are initiated by pressure perturbations and are accelerated by a temperature factor. Furthermore, the prolonged action of low-amplitude ultrasound enables the elongation of amyloid protein nanofibrils directly from natively folded monomeric lysozyme protein, in a controlled manner, until it reaches a critical length. Using solution X-ray scattering, we determined that nanofibrillar assemblies, formed either under the action of sound or from natively fibrillated lysozyme, share identical structural characteristics. Thus, these results provide insights into the effects of ultrasound on fibrillar protein self-assembly and lay the foundation for the potential use of sound energy in protein chemistry.

Error correction is central to many biological systems and is critical for protein function and cell health. During mitosis, error correction is required for the faithful inheritance of genetic material. When functioning properly, the mitotic spindle segregates an equal number of chromosomes to daughter cells with high fidelity. Over the course of spindle assembly, many initially erroneous attachments between kinetochores and microtubules are fixed through the process of error correction. Despite the importance of chromosome segregation errors in cancer and other diseases, there is a lack of methods to characterize the dynamics of error correction and how it can go wrong. Here, we present an experimental method and analysis framework to quantify chromosome segregation error correction in human tissue culture cells with live cell confocal imaging, timed premature anaphase, and automated counting of kinetochores after cell division. We find that errors decrease exponentially over time during spindle assembly. A coarse-grained model, in which errors are corrected in a chromosome-autonomous manner at a constant rate, can quantitatively explain both the measured error correction dynamics and the distribution of anaphase onset times. We further validated our model using perturbations that destabilized microtubules and changed the initial configuration of chromosomal attachments. Taken together, this work provides a quantitative framework for understanding the dynamics of mitotic error correction.

The Luria-Delbrück model is a classic model of population dynamics with random mutations, that has been used historically to prove that random mutations drive evolution. In typical scenarios, the relevant mutation rate is exceedingly small, and mutants are counted only at the final time point. Here, inspired by recent experiments on DNA repair, we study a mathematical model that is formally equivalent to the Luria-Delbrück setup, with the repair rate p playing the role of mutation rate, albeit taking on large values, of order unity per cell division. We find that although at large times the fraction of repaired cells approaches one, the variance of the number of repaired cells undergoes a phase transition: when p>1/2 the variance decreases with time, but, intriguingly, for p

Cell growth and gene expression, essential elements of all living systems, have long been the focus of biophysical interrogation. Advances in single-cell methods have invigorated theoretical studies into these processes. However, until recently, there was little dialog between the two areas of study. Most theoretical models for gene regulation assumed gene activity to be oblivious to the progression of the cell cycle between birth and division. But there are numerous ways in which the periodic character of all cellular observables can modulate gene expression. The molecular factors required for transcription and translation increase in number during the cell cycle, but are also diluted due to the continuous increase in cell volume. The replication of the genome changes the dosage of those same cellular players but also provides competing targets for regulatory binding. Finally, cell division reduces their number again, and so forth. Stochasticity is inherent to all these biological processes, manifested in fluctuations in the synthesis and degradation of new cellular components as well as the random partitioning of molecules at each cell division. The notion of gene expression as stationary is thus hard to justify. In this review, we survey the emerging paradigm of cell-cycle regulated gene expression, with an emphasis on the global expression patterns rather than gene-specific regulation. We discuss recent experimental reports where cell growth and gene expression were simultaneously measured in individual cells, providing first glimpses into the coupling between the two. While the experimental findings, not surprisingly, differ among genes and organisms, several theoretical models have emerged that attempt to reconcile these differences and form a unifying framework for understanding gene expression in growing cells.

In the past decade, great strides have been made to quantify the dynamics of single-cell growth and division in microbes. In order to make sense of the evolutionary history of these organisms, we must understand how features of single-cell growth and division influence evolutionary dynamics. This requires us to connect processes on the single-cell scale to population dynamics. Here, we consider a model of microbial growth in finite populations which explicitly incorporates the single-cell dynamics. We study the behavior of a mutant population in such a model and ask: can the evolutionary dynamics be coarse-grained so that the forces of natural selection and genetic drift can be expressed in terms of the long-term fitness We show that it is in fact not possible, as there is no way to define a single fitness parameter (or reproductive rate) that defines the fate of an organism even in a constant environment. This is due to fluctuations in the population averaged division rate. As a result, various details of the single-cell dynamics affect the fate of a new mutant independently from how they affect the long-term growth rate of the mutant population. In particular, we show that in the case of neutral mutations, variability in generation times increases the rate of genetic drift, and in the case of beneficial mutations, variability decreases its fixation probability. Furthermore, we explain the source of the persistent division rate fluctuations and provide analytic solutions for the fixation probability as a multispecies generalization of the Euler-Lotka equation.

Transcription Factors (TFs) are proteins crucial for regulating gene expression. Effective regulation requires the TFs to rapidly bind to the correct target, enabling the cell to respond timely to various stimuli such as nutrient availability or toxin presence. At the same time, the search process is limited by slow diffusive movement and 'false' targets in DNA - segments of DNA sequences that are similar to the true target. In eukaryotic cells most TFs have an Intrinsically Disordered Region (IDR), which is a long, flexible polymeric tail comprised of hundreds of amino acids. Recent experimental findings indicate that the IDR of certain TFs plays a crucial role in the search process. However, the principles underlying the IDR's role remain unclear. Here, we reveal key design principles of the IDR related to TF binding affinity and search time. Our results indicate a significant enhancement in both of these aspects attributed to the presence of the IDR. Additionally, we propose experiments to verify these theoretical predictions.

The statistics of noise emitted by ultrathin crumpled sheets is measured while they exhibit logarithmic relaxations under load. We find that the logarithmic relaxation advanced via a series of discrete, audible, micromechanical events that are log-Poisson distributed (i.e., the process becomes a Poisson process when time stamps are replaced by their logarithms). The analysis places constraints on the possible mechanisms underlying the glasslike slow relaxation and memory retention in these systems.

How cells regulate their cell cycles is a central question for cell biology. Models of cell size homeostasis have been proposed for bacteria, archaea, yeast, plant, and mammalian cells. New experiments bring forth high volumes of data suitable for testing existing models of cell size regulation and proposing new mechanisms. In this paper, we use conditional independence tests in conjunction with data of cell size at key cell cycle events (birth, initiation of DNA replication, and constriction) in the model bacterium Escherichia coli to select between the competing cell cycle models. We find that in all growth conditions that we study, the division event is controlled by the onset of constriction at midcell. In slow growth, we corroborate a model where replicationrelated processes control the onset of constriction at midcell. In faster growth, we find that the onset of constriction is affected by additional cues beyond DNA replication. Finally, we also find evidence for the presence of additional cues triggering initiations of DNA replication apart from the conventional notion where the mother cells solely determine the initiation event in the daughter cells via an adder per origin model. The use of conditional independence tests is a different approach in the context of understanding cell cycle regulation and it can be used in future studies to further explore the causal links between cell events.

We study the dynamics of flow networks in porous media using two and three dimensional pore-network models. We consider a class of erosion dynamics for a single phase flow with no deposition, chemical reactions, or topology changes assuming a constitutive law depending on flow rate, local velocities, or shear stress at the walls. We show that depending on the erosion law, the flow may become uniform and homogenized or become unstable and develop channels. By defining an order parameter capturing these different behaviors we show that a phase transition occurs depending on the erosion dynamics. Using a simple model, we identify quantitative criteria to distinguish these regimes and correctly predict the fate of the network, and discuss the experimental relevance of our result.

Trinucleotide repeat expansions are responsible for two dozen human disorders. Contracting expanded repeats by Double-Strand Break Repair (DSBR) might be a therapeutic approach. Given the complexity of manipulating human cells, recent assays were made to quantify DSBR efficacy in yeast, using a fluorescent reporter. In this study DSBR is characterized with an interdisciplinary approach, linking large population dynamics and individual cells. Time-resolved molecular measurements of changes in the population are first confronted to a coupled differential equation model to obtain repair processes rates. Comparisons with measurements in microfluidic devices, where the progeny of 80-150 individual cells are followed, show good agreement between individual trajectories and mathematical and molecular results. Further analysis of individual progenies shows the heterogeneity of individual cell contributions to global repair efficacy. Three different categories of repair are identified: high-efficacy error-free, low-efficacy error-free and low-efficacy error-prone. These categories depend on the type of endonuclease used and on the target sequence.

The observation that phenotypic variability is ubiquitous in isogenic populations has led to a multitude of experimental and theoretical studies seeking to probe the causes and consequences of this variability. Whether it be in the context of antibiotic treatments or exponential growth in constant environments, non-genetic variability has significant effects on population dynamics. Here, we review research that elucidates the relationship between cell-to-cell variability and population dynamics. After summarizing the relevant experimental observations, we discuss models of bet-hedging and phenotypic switching. In the context of these models, we discuss how switching between phenotypes at the single-cell level can help populations survive in uncertain environments. Next, we review more fine-grained models of phenotypic variability where the relationship between single-cell growth rates, generation times and cell sizes is explicitly considered. Variability in these traits can have significant effects on the population dynamics, even in a constant environment. We show how these effects can be highly sensitive to the underlying model assumptions. We close by discussing a number of open questions, such as how environmental and intrinsic variability interact and what the role of non-genetic variability in evolutionary dynamics is.

Mechanical rupture, or lysis, of the cytoplasmic membrane is a common cell death pathway in bacteria occurring in response to β-lactam antibiotics. A better understanding of the cellular design principles governing the susceptibility and response of individual cells to lysis could indicate methods of potentiating β-lactam antibiotics and clarify relevant aspects of cellular physiology. Here, we take a single-cell approach to bacterial cell lysis to examine three cellular featuresturgor pressure, mechanosensitive channels, and cell shape changesthat are expected to modulate lysis. We develop a mechanical model of bacterial cell lysis and experimentally analyze the dynamics of lysis in hundreds of single Escherichia coli cells. We find that turgor pressure is the only factor, of these three cellular features, which robustly modulates lysis. We show that mechanosensitive channels do not modulate lysis due to insufficiently fast solute outflow, and that cell shape changes result in more severe cellular lesions but do not influence the dynamics of lysis. These results inform a single-cell view of bacterial cell lysis and underscore approaches of combatting antibiotic tolerance to β-lactams aimed at targeting cellular turgor.

Many unicellular organisms allocate their key proteins asymmetrically between the mother and daughter cells, especially in a stressed environment. A recent theoretical model is able to predict when the asymmetry in segregation of key proteins enhances the population fitness, extrapolating the solution at two limits where the segregation is perfectly asymmetric (asymmetry a = 1) and when the asymmetry is small (0

Gene expression is a stochastic process. Despite the increase of protein numbers in growing cells, the protein concentrations are often found to be confined within small ranges throughout the cell cycle. Generally, the noise in protein concentration can be decomposed into an intrinsic and an extrinsic component, where the former vanishes for high expression levels. Considering the time trajectory of protein concentration as a random walker in the concentration space, an effective restoring force (with a corresponding "spring constant") must exist to prevent the divergence of concentration due to random fluctuations. In this work, we prove that the magnitude of the effective spring constant is directly related to the fraction of intrinsic noise in the total protein concentration noise. We show that one can infer the magnitude of intrinsic, extrinsic, and measurement noises of gene expression solely based on time-resolved data of protein concentration, without any a priori knowledge of the underlying gene expression dynamics. We apply this method to experimental data of single-cell bacterial gene expression. The results allow us to estimate the average copy numbers and the translation burst parameters of the studied proteins.

In biological contexts as diverse as development, apoptosis, and synthetic microbial consortia, collections of cells or subcellular components have been shown to overcome the slow signaling speed of simple diffusion by utilizing diffusive relays, in which the presence of one type of diffusible signaling molecule triggers participation in the emission of the same type of molecule. This collective effect gives rise to fast-traveling diffusive waves. Here, in the context of cell signaling, we show that system dimensionality the shape of the extracellular medium and the distribution of cells within it can dramatically affect the wave dynamics, but that these dynamics are insensitive to details of cellular activation. As an example, we show that neutrophil swarming experiments exhibit dynamical signatures consistent with the proposed signaling motif. We further show that cell signaling relays generate much steeper concentration profiles than does simple diffusion, which may facilitate neutrophil chemotaxis.

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 \u201cmodulated rate\u201d model, in which the clock signal is integrated over time to affect division timing, differs fundamentally from the previously proposed \u201cgating\u201d 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.

We study heat conduction mediated by longitudinal phonons in one-dimensional disordered harmonic chains. Using scaling properties of the phonon density of states and localization in disordered systems, we find nontrivial scaling of the thermal conductance with the system size. Our findings are corroborated by extensive numerical analysis. We show that, suprisingly, the thermal conductance of a system with strong disorder, characterized by a "heavy-tailed" probability distribution, and with large impedance mismatch between the bath and the system, scales normally with the system size, i.e., in a manner consistent with Fourier's law. We identify a dimensionless scaling parameter, related to the temperature scale and the localization length of the phonons, through which the thermal conductance for different models of disorder and different temperatures follows a universal behavior.

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.

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 (\u201cbulging\u201d) after cell wall digestion occurs on a characteristic timescale of 1 s and the growth of the bulge (\u201cswelling\u201d) 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.

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.

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 focused 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 σln(μ)/P remains asymptotically finite whenever accuracy improves from equilibrium, so that μeq/μ→0. Scalings exist, however, even for Hopfield's original network, for which σln(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, σln(μ/μeq)/P

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 LuriaDelbrück 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).

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 (Ho and Amir, 2015). In bacteria, division into two equally sized cells does not broaden the size distribution.

The shapes of most bacteria are imparted by the structures of their peptidoglycan cell walls, which are determined by many dynamic processes that can be described on various length scales ranging from short-range glycan insertions to cellular-scale elasticity. Understanding the mechanisms that maintain stable, rod-like morphologies in certain bacteria has proved to be challenging due to an incomplete understanding of the feedback between growth and the elastic and geometric properties of the cell wall. Here, we probe the effects of mechanical strain on cell shape by modelling the mechanical strains caused by bending and differential growth of the cell wall. We show that the spatial coupling of growth to regions of high mechanical strain can explain the plastic response of cells to bending and quantitatively predict the rate at which bent cells straighten. By growing filamentous Escherichia coli cells in doughnut-shaped microchambers, we find that the cells recovered their straight, native rod-shaped morphologies when released from captivity at a rate consistent with the theoretical prediction. We then measure the localization of MreB, an actin homologue crucial to cell wall synthesis, inside confinement and during the straightening process, and find that it cannot explain the plastic response to bending or the observed straightening rate. Our results implicate mechanical strain sensing, implemented by components of the elongasome yet to be fully characterized, as an important component of robust shape regulation in E. coli.

We observe nonmonotonic aging and memory effects, two hallmarks of glassy dynamics, in two disordered mechanical systems: crumpled thin sheets and elastic foams. Under fixed compression, both systems exhibit monotonic nonexponential relaxation. However, when after a certain waiting time the compression is partially reduced, both systems exhibit a nonmonotonic response: the normal force first increases over many minutes or even hours until reaching a peak value, and only then is relaxation resumed. The peak time scales linearly with the waiting time, indicating that these systems retain long-lasting memory of previous conditions. Our results and the measured scaling relations are in good agreement with a theoretical model recently used to describe observations of monotonic aging in several glassy systems, suggesting that the nonmonotonic behavior may be generic and that athermal systems can show genuine glassy behavior.

Bacteria tightly regulate and coordinate the various events in their cell cycles to duplicate themselves accurately and to control their cell sizes. Growth of Escherichia coli, in particular, follows a relation known as Schaechter's growth law. This law says that the average cell volume scales exponentially with growth rate, with a scaling exponent equal to the time from initiation of a round of DNA replication to the cell division at which the corresponding sister chromosomes segregate. Here, we sought to test the robustness of the growth law to systematic perturbations in cell dimensions achieved by varying the expression levels of mreB and ftsZ. We found that decreasing the mreB level resulted in increased cell width, with little change in cell length, whereas decreasing the ftsZ level resulted in increased cell length. Furthermore, the time from replication termination to cell division increased with the perturbed dimension in both cases. Moreover, the growth law remained valid over a range of growth conditions and dimension perturbations. The growth law can be quantitatively interpreted as a consequence of a tight coupling of cell division to replication initiation. Thus, its robustness to perturbations in cell dimensions strongly supports models in which the timing of replication initiation governs that of cell division, and cell volume is the key phenomenological variable governing the timing of replication initiation. These conclusions are discussed in the context of our recently proposed "adder-per-origin" model, in which cells add a constant volume per origin between initiations and divide a constant time after initiation.

We consider a class of biologically motivated stochastic processes in which a unicellular organism divides its resources (volume or damaged proteins, in particular) symmetrically or asymmetrically between its progeny. Assuming the final amount of the resource is controlled by a growth policy and subject to additive and multiplicative noise, we derive the recursive integral equation describing the evolution of the resource distribution over subsequent generations and use it to study the properties of stable resource distributions. We find conditions under which a unique stable resource distribution exists and calculate its moments for the class of affine linear growth policies. Moreover, we apply an asymptotic analysis to elucidate the conditions under which the stable distribution (when it exists) has a power-law tail. Finally, we use the results of this asymptotic analysis along with the moment equations to draw a stability phase diagram for the system that reveals the counterintuitive result that asymmetry serves to increase stability while at the same time widening the stable distribution. We also briefly discuss how cells can divide damaged proteins asymmetrically between their progeny as a form of damage control. In the appendixes, motivated by the asymmetric division of cell volume in Saccharomyces cerevisiae, we extend our results to the case wherein mother and daughter cells follow different growth policies.

To maintain a constant cell size, dividing cells have to coordinate cell-cycle events with cell growth. This coordination has long been supposed to rely on the existence of size thresholds determining cell-cycle progression [1]. In budding yeast, size is controlled at the G1/S transition [2]. In agreement with this hypothesis, the size at birth influences the time spent in G1: smaller cells have a longer G1 period [3]. Nevertheless, even though cells born smaller have a longer G1, the compensation is imperfect and they still bud at smaller cell sizes. In bacteria, several recent studies have shown that the incremental model of size control, in which size is controlled by addition of a constant volume (in contrast to a size threshold), is able to quantitatively explain the experimental data on four different bacterial species [4-7]. Here, we report on experimental results for the budding yeast Saccharomyces cerevisiae, finding, surprisingly, that cell size control in this organism is very well described by the incremental model, suggesting a common strategy for cell size control with bacteria. Additionally, we argue that for S. cerevisiae the "volume increment" is not added from birth to division, but rather between two budding events.

Characterizing the frequency-dependent response of amorphous systems and glasses can provide important insights into their physics. Here, we study the response of an electron glass, where Coulomb interactions are important and have previously been shown to significantly modify the conductance and lead to memory effects and aging. We propose a model which allows us to take the interactions into account in a self-consistent way, and explore the frequency-dependent conduction at all frequencies. At low frequencies conduction occurs on the percolation backbone, and the model captures the variable-range-hopping behavior. At high frequencies conduction is dominated by localized clusters. Despite the difference in physical mechanisms at low and high frequency, we are able to approximately scale all numerical data onto a single curve, using two parameters: the DC conduction and the DC dielectric constant. The behavior follows the universal scaling that is experimentally observed for a large class of amorphous solids.

Cell walls define a cell's shape in bacteria. The walls are rigid to resist large internal pressures, but remarkably plastic to adapt to a wide range of external forces and geometric constraints. Currently, it is unknown how bacteria maintain their shape. In this paper, we develop experimental and theoretical approaches and show that mechanical stresses regulate bacterial cell wall growth. By applying a precisely controllable hydrodynamic force to growing rod-shaped Escherichia coli and Bacillus subtilis cells, we demonstrate that the cells can exhibit two fundamentally different modes of deformation. The cells behave like elastic rods when subjected to transient forces, but deform plastically when significant cell wall synthesis occurs while the force is applied. The deformed cells always recover their shape. The experimental results are in quantitative agreementwith the predictions of the theory of dislocation-mediated growth. In particular, we find that a single dimensionless parameter, which depends on a combination of independently measured physical properties of the cell, can describe the cell's responses under various experimental conditions. These findings provide insight into how living cells robustly maintain their shape under varying physical environments.

Interference is the source of some of the spectacular colors of animals and plants in nature. In some of these systems, the physical structure consists of an ordered array of layers with alternating high and low refractive indices. This periodicity leads to an optical band structure that is analogous to the electronic band structure encountered in semiconductor physics: specific bands of wavelengths (the stop bands) are perfectly reflected. Here, we present a minimal model for optical band structure in a periodic multilayer structure and solve it using recursion relations. The stop bands emerge in the limit of an infinite number of layers by finding the fixed point of the recursion. We compare to experimental data for various beetles, whose optical structure resembles the proposed model. Thus, using only the phenomenon of interference and the idea of recursion, we are able to elucidate the concept of band structure in the context of the experimentally observed high reflectance and iridescent appearance of structurally colored beetles.

There is a deep analogy between the physics of crystalline solids and the behavior of superfluids, dating back to pioneering work of Phillip Anderson, Paul Martin and others. The stiffness to shear deformations in a periodic crystal resembles the superfluid density that controls the behavior of supercurrents in neutral superfluids such as He^4. Dislocations in solids have a close analogy with quantized vortices in superfluids. Remarkable recent experiments on the way rod-shaped bacteria elongate their cell walls have focused attention on the dynamics and interactions of point-like dislocation defects in partially ordered cylindrical crystalline monolayers. In these lectures, we review the physics of superfluid helium films on cylinders and discuss how confinement in one direction affects vortex interactions with supercurrents. Although there are similarities with the way dislocations respond to strains on cylinders, important differences emerge, due to the vector nature of the topological charges characterizing the dislocations.

We study the spectral properties of a class of random matrices where the matrix elements depend exponentially on the distance between uniformly and randomly distributed points. This model arises naturally in various physical contexts, such as the diffusion of particles, slow relaxations in glasses, and scalar phonon localization. Using a combination of a renormalization group procedure and a direct moment calculation, we find the eigenvalue distribution density (i.e., the spectrum), for low densities, and the localization properties of the eigenmodes, for arbitrary dimension. Finally, we discuss the physical implications of the results.

Glassy systems are ubiquitous in nature. They are characterized by slow relaxations to equilibrium without a typical time scale, aging, and memory effects. Understanding this has been a long-standing problem in physics. We study the aging of the electron glass, a system showing remarkable slow relaxations of the conductance. We find that the appropriate broad distribution of relaxation rates leads to a universal relaxation of the form log(1+tw/t) for the common aging protocol, where tw is the length of time the perturbation driving the system out of equilibrium was on, and t the time of measurement. These results agree well with several experiments performed on different glassy systems, and examining different physical observables, for times ranging from seconds to several hours. The suggested theoretical framework appears to offer a paradigm for aging in a broad class of glassy materials.