Dynamics of protein distributions in cell populations

Coordination of gene expression noise with cell size: analytical results for agent-based models of growing cell populations

The chemical master equation and the Gillespie algorithm are widely used to model the reaction kinetics inside living cells. It is thereby assumed that cell growth and division can be modelled through effective dilution reactions and extrinsic noise sources. We here re-examine these paradigms through developing an analytical agent-based framework of growing and dividing cells accompanied by an exact simulation algorithm, which allows us to quantify the dynamics of virtually any intracellular reaction network affected by stochastic cell size control and division noise. We find that the solution of the chemical master equation-including static extrinsic noise-exactly agrees with the agent-based formulation when the network under study exhibits stochastic concentration homeostasis, a novel condition that generalizes concentration homeostasis in deterministic systems to higher order moments and distributions. We illustrate stochastic concentration homeostasis for a range of common gene expression networks. When this condition is not met, we demonstrate by extending the linear noise approximation to agent-based models that the dependence of gene expression noise on cell size can qualitatively deviate from the chemical master equation. Surprisingly, the total noise of the agent-based approach can still be well approximated by extrinsic noise models.

Stochastic modeling of cell growth with symmetric or asymmetric division

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.

Lineage Tracking for Probing Heritable Phenotypes at Single-Cell Resolution

Determining the phenotype and genotype of single cells is central to understand microbial evolution. DNA sequencing technologies allow the detection of mutants at high resolution, but similar approaches for phenotypic analyses are still lacking. We show that a drop-based millifluidic system enables the detection of heritable phenotypic changes in evolving bacterial populations. At time intervals, cells were sampled and individually compartmentalized in 100 nL drops. Growth through 15 generations was monitored using a fluorescent protein reporter. Amplification of heritable changes–via growth–over multiple generations yields phenotypically distinct clusters reflecting variation relevant for evolution. To demonstrate the utility of this approach, we follow the evolution of Escherichia coli populations during 30 days of starvation. Phenotypic diversity was observed to rapidly increase upon starvation with the emergence of heritable phenotypes. Mutations corresponding to each phenotypic class were identified by DNA sequencing. This scalable lineage-tracking technology opens the door to large-scale phenotyping methods with special utility for microbiology and microbial population biology.

Universal protein distributions in a model of cell growth and division

Protein distributions measured under a broad set of conditions in bacteria and yeast were shown to exhibit a common skewed shape, with variances depending quadratically on means. For bacteria these properties were reproduced by temporal measurements of protein content, showing accumulation and division across generations. Here we present a stochastic growth-and-division model with feedback which captures these observed properties. The limiting copy number distribution is calculated exactly, and a single parameter is found to determine the distribution shape and the variance-to-mean relation. Estimating this parameter from bacterial temporal data reproduces the measured distribution shape with high accuracy and leads to predictions for future experiments.

Single-cell protein dynamics reproduce universal fluctuations in cell populations

Protein variability in single cells has been studied extensively in populations, but little is known about temporal protein fluctuations in a single cell over extended times. We present here traces of protein copy number measured in individual bacteria over multiple generations and investigate their statistical properties, comparing them to previously measured population snapshots. We find that temporal fluctuations in individual cells exhibit the same properties as those previously observed in populations. Scaled fluctuations around the mean of each trace exhibit the universal distribution shape measured in populations under a wide range of conditions and in two distinct microorganisms; the mean and variance of the traces over time obey the same quadratic relation. Analyzing the individual protein traces reveals that within a cell cycle protein content increases exponentially, with a rate that varies from cycle to cycle. This leads to a compact description of the trace as a 3-variable stochastic process -exponential rate, cell cycle duration and value at the cycle start- sampled once a cycle. This description is sufficient to reproduce both universal statistical properties of the protein fluctuations. Our results show that the protein distribution shape is insensitive to sub-cycle intracellular microscopic details and reflects global cellular properties that fluctuate between generations.

The unforeseen challenge: from genotype-to-phenotype in cell populations

Biological cells present a paradox, in that they show simultaneous stability and flexibility, allowing them to adapt to new environments and to evolve over time. The emergence of stable cell states depends on genotype-to-phenotype associations, which essentially reflect the organization of gene regulatory modes. The view taken here is that cell-state organization is a dynamical process in which the molecular disorder manifests itself in a macroscopic order. The genome does not determine the ordered cell state; rather, it participates in this process by providing a set of constraints on the spectrum of regulatory modes, analogous to boundary conditions in physical dynamical systems. We have developed an experimental framework, in which cell populations are exposed to unforeseen challenges; novel perturbations they had not encountered before along their evolutionary history. This approach allows an unbiased view of cell dynamics, uncovering the potential of cells to evolve and develop adapted stable states. In the last decade, our experiments have revealed a coherent set of observations within this framework, painting a picture of the living cell that in many ways is not aligned with the conventional one. Of particular importance here, is our finding that adaptation of cell-state organization is essentially an efficient exploratory dynamical process rather than one founded on random mutations. Based on our framework, a set of concepts underlying cell-state organization-exploration evolving by global, non-specific, dynamics of gene activity-is presented here. These concepts have significant consequences for our understanding of the emergence and stabilization of a cell phenotype in diverse biological contexts. Their implications are discussed for three major areas of biological inquiry: evolution, cell differentiation and cancer. There is currently no unified theoretical framework encompassing the emergence of order, a stable state, in the living cell. Hopefully, the integrated picture described here will provide a modest contribution towards a physics theory of the cell.

Weak correlation of starch and volume in synchronized photosynthetic cells

In cultures of unicellular algae, features of single cells, such as cellular volume and starch content, are thought to be the result of carefully balanced growth and division processes. Single-cell analyses of synchronized photoautotrophic cultures of the unicellular alga Chlamydomonas reinhardtii reveal, however, that the cellular volume and starch content are only weakly correlated. Likewise, other cell parameters, e.g., the chlorophyll content per cell, are only weakly correlated with cell size. We derive the cell size distributions at the beginning of each synchronization cycle considering growth, timing of cell division and daughter cell release, and the uneven division of cell volume. Furthermore, we investigate the link between cell volume growth and starch accumulation. This work presents evidence that, under the experimental conditions of light-dark synchronized cultures, the weak correlation between both cell features is a result of a cumulative process rather than due to asymmetric partition of biomolecules during cell division. This cumulative process necessarily limits cellular similarities within a synchronized cell population.

Synaptic size dynamics as an effectively stochastic process

Long-term, repeated measurements of individual synaptic properties have revealed that synapses can undergo significant directed and spontaneous changes over time scales of minutes to weeks. These changes are presumably driven by a large number of activity-dependent and independent molecular processes, yet how these processes integrate to determine the totality of synaptic size remains unknown. Here we propose, as an alternative to detailed, mechanistic descriptions, a statistical approach to synaptic size dynamics. The basic premise of this approach is that the integrated outcome of the myriad of processes that drive synaptic size dynamics are effectively described as a combination of multiplicative and additive processes, both of which are stochastic and taken from distributions parametrically affected by physiological signals. We show that this seemingly simple model, known in probability theory as the Kesten process, can generate rich dynamics which are qualitatively similar to the dynamics of individual glutamatergic synapses recorded in long-term time-lapse experiments in ex-vivo cortical networks. Moreover, we show that this stochastic model, which is insensitive to many of its underlying details, quantitatively captures the distributions of synaptic sizes measured in these experiments, the long-term stability of such distributions and their scaling in response to pharmacological manipulations. Finally, we show that the average kinetics of new postsynaptic density formation measured in such experiments is also faithfully captured by the same model. The model thus provides a useful framework for characterizing synapse size dynamics at steady state, during initial formation of such steady states, and during their convergence to new steady states following perturbations. These findings show the strength of a simple low dimensional statistical model to quantitatively describe synapse size dynamics as the integrated result of many underlying complex processes.

Simulating cortical development as a self constructing process: a novel multi-scale approach combining molecular and physical aspects

Current models of embryological development focus on intracellular processes such as gene expression and protein networks, rather than on the complex relationship between subcellular processes and the collective cellular organization these processes support. We have explored this collective behavior in the context of neocortical development, by modeling the expansion of a small number of progenitor cells into a laminated cortex with layer and cell type specific projections. The developmental process is steered by a formal language analogous to genomic instructions, and takes place in a physically realistic three-dimensional environment. A common genome inserted into individual cells control their individual behaviors, and thereby gives rise to collective developmental sequences in a biologically plausible manner. The simulation begins with a single progenitor cell containing the artificial genome. This progenitor then gives rise through a lineage of offspring to distinct populations of neuronal precursors that migrate to form the cortical laminae. The precursors differentiate by extending dendrites and axons, which reproduce the experimentally determined branching patterns of a number of different neuronal cell types observed in the cat visual cortex. This result is the first comprehensive demonstration of the principles of self-construction whereby the cortical architecture develops. In addition, our model makes several testable predictions concerning cell migration and branching mechanisms.

The proper operation of the brain depends on the correct developmental wiring of billions of neurons. Understanding this process of living self-construction is crucial not only for biological explanation and medical therapy, but could also provide an entirely new approach to industrial fabrication. We are approaching this problem through detailed simulation of cortical development. We have previously presented a software package that allows for simulation of cellular growth in a 3D space that respects physical forces and diffusion of substances, as well as an instruction language for specifying biologically plausible ‘genetic codes’. Here we apply this novel formalism to understanding the principles of cortical development in the context of multiple, spatially distributed agents that communicate only by local metabolic messages.

Metabolic variability in micro-populations

Biological cells in a population are variable in practically every property. Much is known about how variability of single cells is reflected in the statistical properties of infinitely large populations; however, many biologically relevant situations entail finite times and intermediate-sized populations. The statistical properties of an ensemble of finite populations then come into focus, raising questions concerning inter-population variability and dependence on initial conditions. Recent technologies of microfluidic and microdroplet-based population growth realize these situations and make them immediately relevant for experiments and biotechnological application. We here study the statistical properties, arising from metabolic variability of single cells, in an ensemble of micro-populations grown to saturation in a finite environment such as a micro-droplet. We develop a discrete stochastic model for this growth process, describing the possible histories as a random walk in a phenotypic space with an absorbing boundary. Using a mapping to Polya’s Urn, a classic problem of probability theory, we find that distributions approach a limiting inoculum-dependent form after a large number of divisions. Thus, population size and structure are random variables whose mean, variance and in general their distribution can reflect initial conditions after many generations of growth. Implications of our results to experiments and to biotechnology are discussed.

Universal protein fluctuations in populations of microorganisms

The copy number of any protein fluctuates among cells in a population; characterizing and understanding these fluctuations is a fundamental problem in biophysics. We show here that protein distributions measured under a broad range of biological realizations collapse to a single non-gaussian curve under scaling by the first two moments. Moreover, in all experiments the variance is found to depend quadratically on the mean, showing that a single degree of freedom determines the entire distribution. Our results imply that protein fluctuations do not reflect any specific molecular or cellular mechanism, and suggest that some buffering process masks these details and induces universality.

Random partitioning of molecules at cell division

Many RNAs, proteins, and organelles are present in such low numbers per cell that random segregation of individual copies causes large "partitioning errors" at cell division. Even symmetrically dividing cells can then by chance produce daughters with very different composition. The size of the errors depends on the segregation mechanism: Control systems can reduce low-abundance errors, but the segregation process can also be subject to upstream sources of randomness or spatial heterogeneities that create large errors despite high abundances. Here we mathematically demonstrate how partitioning errors arise for different types of segregation mechanisms and how errors can be greatly increased by upstream heterogeneity but remarkably hard to avoid through controlled partitioning. We also show that seemingly straightforward experiments cannot be straightforwardly interpreted because very different mechanisms produce identical fits and present an approach to deal with this problem by adding binomial counting noise and testing for convexity or concavity in the partitioning error as a function of the binomial thinning parameter. The results lay a conceptual groundwork for more effective studies of heterogeneity among growing and dividing cells, whether in microbes or in differentiating tissues.

Collective dynamics of gene expression in cell populations

The phenotypic state of the cell is commonly thought to be determined by the set of expressed genes. However, given the apparent complexity of genetic networks, it remains open what processes stabilize a particular phenotypic state. Moreover, it is not clear how unique is the mapping between the vector of expressed genes and the cell's phenotypic state. To gain insight on these issues, we study here the expression dynamics of metabolically essential genes in twin cell populations. We show that two yeast cell populations derived from a single steady-state mother population and exhibiting a similar growth phenotype in response to an environmental challenge, displayed diverse expression patterns of essential genes. The observed diversity in the mean expression between populations could not result from stochastic cell-to-cell variability, which would be averaged out in our large cell populations. Remarkably, within a population, sets of expressed genes exhibited coherent dynamics over many generations. Thus, the emerging gene expression patterns resulted from collective population dynamics. It suggests that in a wide range of biological contexts, gene expression reflects a self-organization process coupled to population-environment dynamics.

Intraclonal protein expression heterogeneity in recombinant CHO cells

Therapeutic glycoproteins have played a major role in the commercial success of biotechnology in the post-genomic era. But isolating recombinant mammalian cell lines for large-scale production remains costly and time-consuming, due to substantial variation and unpredictable stability of expression amongst transfected cells, requiring extensive clone screening to identify suitable high producers. Streamlining this process is of considerable interest to industry yet the underlying phenomena are still not well understood. Here we examine an antibody-expressing Chinese hamster ovary (CHO) clone at single-cell resolution using flow cytometry and vectors, which couple light and heavy chain transcription to fluorescent markers. Expression variation has traditionally been attributed to genetic heterogeneity arising from random genomic integration of vector DNA. It follows that single cell cloning should yield a homogeneous cell population. We show, in fact, that expression in a clone can be surprisingly heterogeneous (standard deviation 50 to 70% of the mean), approaching the level of variation in mixed transfectant pools, and each antibody chain varies in tandem. Phenotypic variation is fully developed within just 18 days of cloning, yet is not entirely explained by measurement noise, cell size, or the cell cycle. By monitoring the dynamic response of subpopulations and subclones, we show that cells also undergo slow stochastic fluctuations in expression (half-life 2 to 11 generations). Non-genetic diversity may therefore play a greater role in clonal variation than previously thought. This also has unexpected implications for expression stability. Stochastic gene expression noise and selection bias lead to perturbations from steady state at the time of cloning. The resulting transient response as clones reestablish their expression distribution is not ordinarily accounted for but can contribute to declines in median expression over timescales of up to 50 days. Noise minimization may therefore be a novel strategy to reduce apparent expression instability and simplify cell line selection.

Model-based deconvolution of cell cycle time-series data reveals gene expression details at high resolution

In both prokaryotic and eukaryotic cells, gene expression is regulated across the cell cycle to ensure “just-in-time” assembly of select cellular structures and molecular machines. However, present in all time-series gene expression measurements is variability that arises from both systematic error in the cell synchrony process and variance in the timing of cell division at the level of the single cell. Thus, gene or protein expression data collected from a population of synchronized cells is an inaccurate measure of what occurs in the average single-cell across a cell cycle. Here, we present a general computational method to extract “single-cell”-like information from population-level time-series expression data. This method removes the effects of 1) variance in growth rate and 2) variance in the physiological and developmental state of the cell. Moreover, this method represents an advance in the deconvolution of molecular expression data in its flexibility, minimal assumptions, and the use of a cross-validation analysis to determine the appropriate level of regularization. Applying our deconvolution algorithm to cell cycle gene expression data from the dimorphic bacterium Caulobacter crescentus, we recovered critical features of cell cycle regulation in essential genes, including ctrA and ftsZ, that were obscured in population-based measurements. In doing so, we highlight the problem with using population data alone to decipher cellular regulatory mechanisms and demonstrate how our deconvolution algorithm can be applied to produce a more realistic picture of temporal regulation in a cell.

Time-series analyses of cellular regulatory processes have successfully drawn attention to the importance of temporal regulation in biological systems. A number of model systems can be synchronized such that data collected on cell populations better reflect the dynamic properties of the individual cell. However, experimental synchronization is never perfect, and the degree of synchrony that does exist at the outset of an experiment is quickly lost over time as cells grow at different rates and enter different developmental or physiological states on cell division. Thus, data collected from a population of synchronized cells can lead to incorrect models of temporal regulation. Here we demonstrate that the problem of relating population data to the individual cell can be resolved with a computational method that effectively removes the effects of both imperfect synchrony and time-dependent loss of synchrony. Application of this deconvolution algorithm to a cell cycle time-series data set from the model bacterium Caulobacter crescentus uncovers critical temporal details in the expression of essential genes that are not evident in the raw population-based data. The deconvolution routine presented here is a robust and general tool for extracting biochemical parameters of the average single cell from population time-series data.

A novel assay based on fluorescence microscopy and image processing for determining phenotypic distributions of rod-shaped bacteria

Cell population balance (CPB) models can account for the phenotypic heterogeneity that characterizes isogenic cell populations. To utilize the predictive power of these models, however, we must determine the single-cell reaction and division rates as well as the partition probability density function of the cell population. These functions can be obtained through the Collins-Richmond inverse CPB modeling methodology, if we know the phenotypic distributions of (a) the overall cell population, (b) the dividing cell subpopulation, and (c) the newborn cell subpopulation. This study presents the development of a novel assay that combines fluorescence microscopy and image processing to determine these distributions. The method is generally applicable to rod-shaped cells dividing through the formation of a characteristic constriction. Morphological criteria were developed for the automatic identification of dividing cells and validated through direct comparison with manually obtained measurements. The newborn cell subpopulation was obtained from the corresponding dividing cell subpopulation by collecting information from the two compartments separated by the constriction. The method was applied to E. coli cells carrying the genetic toggle network with a green fluorescent marker. Our measurements for the overall cell population were in excellent agreement with the distributions obtained via flow cytometry. The new assay constitutes a powerful tool that can be used in conjunction with inverse CPB modeling to rigorously quantify single-cell behavior from data collected from highly heterogeneous cell populations.

The importance of being persistent: heterogeneity of bacterial populations under antibiotic stress

While the DNA sequence is largely responsible for transmitting phenotypic traits over evolutionary time, organisms are also considerably affected by phenotypic variations that persist for more than one generation, with no direct change in the organisms' DNA sequence. In contrast to genetic variation, which is passed on over many generations, the phenotypic variation generated by nongenetic mechanisms is difficult to study due to the inherently limited life time of states that are not encoded in the DNA sequence, but makes it possible for the 'memory' of past environments to influence future organisms. One striking example of phenotypic variation is the phenomenon of bacterial persistence, whereby genetically identical bacterial populations respond heterogeneously to antibiotic treatment. Our aim is to review several experimental and theoretical approaches to the study of persistence. We define persistence as a characteristic of a heterogeneous bacterial population that is taken as a generic example through which we illustrate the approach and study the dynamics of population variability. The clinical and evolutionary implications of persistence are discussed in light of the mathematical description. This approach should be of relevance to the study of other phenomena in which nongenetic variability is involved, such as cellular differentiation or the response of cancer cells to treatment.

Cellular properties and population asymptotics in the population balance equation

Proliferating cell populations at steady-state growth often exhibit broad protein distributions with exponential tails. The sources of this variation and its universality are of much theoretical interest. Here we address the problem by asymptotic analysis of the population balance equation. We show that the steady-state distribution tail is determined by a combination of protein production and cell division and is insensitive to other model details. Under general conditions this tail is exponential with a dependence on parameters consistent with experiment. We discuss the conditions for this effect to be dominant over other sources of variation and the relation to experiments.

Nonequilibrium statistical mechanics of dividing cell populations

We present and study a model for the nonequilibrium statistical mechanics of protein distributions in a proliferating cell population. Our model describes how the total protein variation is shaped by two processes: variation in protein production internal to the cells and variation in division and inheritance at the population level. It enables us to assess the contribution of each of these components separately. We find that, even if production is deterministic, cell division can generate a large variation in protein distribution. In this limit we solve exactly a special case and draw an analogy between protein distribution along cell generations and stress distribution in layers of granular material. At the other limit of extremely noisy protein production, we find that the population structure restrains variation and that the details of division do not affect the tail of the distribution.