Contingency and statistical laws in replicate microbial closed ecosystems.

Contingency, the persistent influence of past random events, pervades biology. To what extent, then, is each course of ecological or evolutionary dynamics unique, and to what extent are these dynamics subject to a common statistical structure? Addressing this question requires replicate measurements to search for emergent statistical laws. We establish a readily replicated microbial closed ecosystem (CES), sustaining its three species for years. We precisely measure the local population density of each species in many CES replicates, started from the same initial conditions and kept under constant light and temperature. The covariation among replicates of the three species densities acquires a stable structure, which could be decomposed into discrete eigenvectors, or "ecomodes." The largest ecomode dominates population density fluctuations around the replicate-average dynamics. These fluctuations follow simple power laws consistent with a geometric random walk. Thus, variability in ecological dynamics can be studied with CES replicates and described by simple statistical laws.

A theoretical perspective on Waddington's genetic assimilation experiments.

Genetic assimilation is the process by which a phenotype that is initially induced by an environmental stimulus becomes stably inherited in the absence of the stimulus after a few generations of selection. While the concept has attracted much debate after being introduced by C. H. Waddington 70 y ago, there have been few experiments to quantitatively characterize the phenomenon. Here, we revisit and organize the results of Waddington's original experiments and follow-up studies that attempted to replicate his results. We then present a theoretical model to illustrate the process of genetic assimilation and highlight several aspects that we think require further quantitative studies, including the gradual increase of penetrance, the statistics of delay in assimilation, and the frequency of unviability during selection. Our model captures Waddington's picture of developmental paths in a canalized landscape using a stochastic dynamical system with alternative trajectories that can be controlled by either external signals or internal variables. It also reconciles two descriptions of the phenomenon-Waddington's, expressed in terms of an individual organism's developmental paths, and that of Bateman in terms of the population distribution crossing a hypothetical threshold. Our results provide theoretical insight into the concepts of canalization, phenotypic plasticity, and genetic assimilation.

Evolutionary dynamics, evolutionary forces, and robustness: A nonequilibrium statistical mechanics perspective.

SignificanceEvolution through natural selection is an overwhelmingly complex process, and it is not surprising that theoretical approaches are strongly simplifying it. For instance, population genetics considers mainly dynamics of gene allele frequencies. Here, we develop a complementary approach to evolutionary dynamics based on three elements-organism reproduction, variations, and selection-that are essential for any evolutionary theory. By considering such general dynamics as a stochastic thermodynamic process, we clarify the nature and action of the evolutionary forces. We show that some of the forces cannot be described solely in terms of fitness landscapes. We also find that one force contribution can make organism reproduction insensitive (robust) to variations.

Protein sectors: evolutionary units of three-dimensional structure.

Proteins display a hierarchy of structural features at primary, secondary, tertiary, and higher-order levels, an organization that guides our current understanding of their biological properties and evolutionary origins. Here, we reveal a structural organization distinct from this traditional hierarchy by statistical analysis of correlated evolution between amino acids. Applied to the S1A serine proteases, the analysis indicates a decomposition of the protein into three quasi-independent groups of correlated amino acids that we term "protein sectors." Each sector is physically connected in the tertiary structure, has a distinct functional role, and constitutes an independent mode of sequence divergence in the protein family. Functionally relevant sectors are evident in other protein families as well, suggesting that they may be general features of proteins. We propose that sectors represent a structural organization of proteins that reflects their evolutionary histories.

Nongenetic individuality, changeability, and inheritance in bacterial behavior.

Isogenic populations often display remarkable levels of phenotypic diversity even in constant, homogeneous environments. Such diversity results from differences between individuals ("nongenetic individuality") as well as changes during individuals' lifetimes ("changeability"). Yet, studies that capture and quantify both sources of diversity are scarce. Here we measure the swimming behavior of hundreds of Escherichia coli bacteria continuously over two generations and use a model-independent method for quantifying behavior to show that the behavioral space of E. coli is low-dimensional, with variations occurring mainly along two independent and interpretable behavioral traits. By statistically decomposing the diversity in these two traits, we find that individuality is the main source of diversity, while changeability makes a smaller but significant contribution. Finally, we show that even though traits of closely related individuals can be remarkably different, they exhibit positive correlations across generations that imply nongenetic inheritance. The model-independent experimental and theoretical framework developed here paves the way for more general studies of microbial behavioral diversity.

Lessons from equilibrium statistical physics regarding the assembly of protein complexes.

Cellular functions are established through biological evolution, but are constrained by the laws of physics. For instance, the physics of protein folding limits the lengths of cellular polypeptide chains. Consequently, many cellular functions are carried out not by long, isolated proteins, but rather by multiprotein complexes. Protein complexes themselves do not escape physical constraints, one of the most important being the difficulty of assembling reliably in the presence of cellular noise. In order to lay the foundation for a theory of reliable protein complex assembly, we study here an equilibrium thermodynamic model of self-assembly that exhibits 4 distinct assembly behaviors: diluted protein solution, liquid mixture, "chimeric assembly," and "multifarious assembly." In the latter regime, different protein complexes can coexist without forming erroneous chimeric structures. We show that 2 conditions have to be fulfilled to attain this regime: 1) The composition of the complexes needs to be sufficiently heterogeneous, and 2) the use of the set of components by the complexes has to be sparse. Our analysis of publicly available databases of protein complexes indicates that cellular protein systems might have indeed evolved so as to satisfy both of these conditions.

Environment-to-phenotype mapping and adaptation strategies in varying environments.

Biological organisms exhibit diverse strategies for adapting to varying environments. For example, a population of organisms may express the same phenotype in all environments ("unvarying strategy") or follow environmental cues and express alternative phenotypes to match the environment ("tracking strategy"), or diversify into coexisting phenotypes to cope with environmental uncertainty ("bet-hedging strategy"). We introduce a general framework for studying how organisms respond to environmental variations, which models an adaptation strategy by an abstract mapping from environmental cues to phenotypic traits. Depending on the accuracy of environmental cues and the strength of natural selection, we find different adaptation strategies represented by mappings that maximize the long-term growth rate of a population. The previously studied strategies emerge as special cases of our model: The tracking strategy is favorable when environmental cues are accurate, whereas when cues are noisy, organisms can either use an unvarying strategy or, remarkably, use the uninformative cue as a source of randomness to bet hedge. Our model of the environment-to-phenotype mapping is based on a network with hidden units; the performance of the strategies is shown to rely on having a high-dimensional internal representation, which can even be random.

Homeorhesis and ecological succession quantified in synthetic microbial ecosystems.

The dynamics of ecological change following a major perturbation, known as succession, are influenced by random processes. Direct quantitation of the degree of contingency in succession requires chronological study of replicate ecosystems. We previously found that population dynamics in carefully controlled, replicated synthetic microbial ecosystems were strongly deterministic over several months. Here, we present simplified, two-species microbial ecosystems consisting of algae and ciliates, imaged in toto at single-cell resolution with fluorescence microscopy over a period of 1 to 2 weeks. To directly study succession in these ecosystems, we deliberately varied the initial cell abundances over replicates and quantified the ensuing dynamics. The distribution of abundance trajectories rapidly converged to a nearly deterministic path, with small fluctuations, despite variations in initial conditions, environmental perturbations, and intrinsic noise, indicative of homeorhesis. Homeorhesis was also observed for certain phenotypic variables, such as partitioning of the ciliates into distinct size classes and clumping of the algae. Although the mechanism of homeorhesis observed in these synthetic ecosystems remains to be elucidated, it is clear that it must emerge from the ways each species controls its own internal states, with respect to a diverse set of environmental conditions and ecological interactions.

Benefits of phenotypic plasticity for population growth in varying environments.

Phenotypic plasticity refers to the capacity of the same organisms to exhibit different characteristics under varied environmental conditions. A plastic developmental program allows organisms to sense environmental cues in early stages of life and express phenotypes that are better fitted to environments encountered later in life. This is often considered an adaptive strategy for living in varying environments as long as the plastic response is sufficiently fast, is accurate, and is not too costly. However, despite direct costs of maintaining plasticity and producing phenotypes, a fundamental constraint on the benefit of phenotypic plasticity comes from the predictability of the future environment based on the environmental cues received during development. Here, we analyze a model of plastic development and derive the limits within which this strategy can promote population growth. An explicit expression for the long-term growth rate of a developmentally plastic population is found, which can be decomposed into several easily interpretable terms, representing the benefits and the limitations of phenotypic plasticity as an adaptation strategy. This growth rate decomposition has a remarkably similar form to the expressions previously obtained for the bet-hedging strategy, in which a population randomly diversifies into coexisting subgroups with different phenotypes, implying that those evolutionary strategies may be unified under a common general framework.

Evolutionary learning of adaptation to varying environments through a transgenerational feedback.

Organisms can adapt to a randomly varying environment by creating phenotypic diversity in their population, a phenomenon often referred to as "bet hedging." The favorable level of phenotypic diversity depends on the statistics of environmental variations over timescales of many generations. Could organisms gather such long-term environmental information to adjust their phenotypic diversity? We show that this process can be achieved through a simple and general learning mechanism based on a transgenerational feedback: The phenotype of the parent is progressively reinforced in the distribution of phenotypes among the offspring. The molecular basis of this learning mechanism could be searched for in model organisms showing epigenetic inheritance.

Strain analysis of protein structures and low dimensionality of mechanical allosteric couplings.

In many proteins, especially allosteric proteins that communicate regulatory states from allosteric to active sites, structural deformations are functionally important. To understand these deformations, dynamical experiments are ideal but challenging. Using static structural information, although more limited than dynamical analysis, is much more accessible. Underused for protein analysis, strain is the natural quantity for studying local deformations. We calculate strain tensor fields for proteins deformed by ligands or thermal fluctuations using crystal and NMR structure ensembles. Strains-primarily shears-show deformations around binding sites. These deformations can be induced solely by ligand binding at distant allosteric sites. Shears reveal quasi-2D paths of mechanical coupling between allosteric and active sites that may constitute a widespread mechanism of allostery. We argue that strain-particularly shear-is the most appropriate quantity for analysis of local protein deformations. This analysis can reveal mechanical and biological properties of many proteins.

Elastic strain and twist analysis of protein structural data and allostery of the transmembrane channel KcsA.

The abundance of available static protein structural data makes the more effective analysis and interpretation of this data a valuable tool to supplement the experimental study of protein mechanics. Structural displacements can be difficult to analyze and interpret. Previously, we showed that strains provide a more natural and interpretable representation of protein deformations, revealing mechanical coupling between spatially distinct sites of allosteric proteins. Here, we demonstrate that other transformations of displacements yield additional insights. We calculate the divergence and curl of deformations of the transmembrane channel KcsA. Additionally, we introduce quantities analogous to bend, splay, and twist deformation energies of nematic liquid crystals. These transformations enable the decomposition of displacements into different modes of deformation, helping to characterize the type of deformation a protein undergoes. We apply these calculations to study the filter and gating regions of KcsA. We observe a continuous path of rotational deformations physically coupling these two regions, and, we propose, underlying the allosteric interaction between these regions. Bend, splay, and twist distinguish KcsA gate opening, filter opening, and filter-gate coupling, respectively. In general, physically meaningful representations of deformations (like strain, curl, bend, splay, and twist) can make testable predictions and yield insights into protein mechanics, augmenting experimental methods and more fully exploiting available structural data.

Multifarious assembly mixtures: systems allowing retrieval of diverse stored structures.

Self-assembly materials are traditionally designed so that molecular or mesoscale components form a single kind of large structure. Here, we propose a scheme to create "multifarious assembly mixtures," which self-assemble many different large structures from a set of shared components. We show that the number of multifarious structures stored in the solution of components increases rapidly with the number of different types of components. However, each stored structure can be retrieved by tuning only a few parameters, the number of which is only weakly dependent on the size of the assembled structure. Implications for artificial and biological self-assembly are discussed.

Bet Hedging against Demographic Fluctuations.

Biological organisms have to cope with stochastic variations in both the external environment and the internal population dynamics. Theoretical studies and laboratory experiments suggest that population diversification could be an effective bet-hedging strategy for adaptation to varying environments. Here we show that bet hedging can also be effective against demographic fluctuations that pose a trade-off between growth and survival for populations even in a constant environment. A species can maximize its overall abundance in the long term by diversifying into coexisting subpopulations of both "fast-growing" and "better-surviving" individuals. Our model generalizes statistical physics models of birth-death processes to incorporate dispersal, during which new populations are founded, and can further incorporate variations of local environments. In this way, we unify different bet-hedging strategies against demographic and environmental variations as a general means of adaptation to both types of uncertainties in population growth.

A model for the generation and transmission of variations in evolution.

The inheritance of characteristics induced by the environment has often been opposed to the theory of evolution by natural selection. However, although evolution by natural selection requires new heritable traits to be produced and transmitted, it does not prescribe, per se, the mechanisms by which this is operated. The mechanisms of inheritance are not, however, unconstrained, because they are themselves subject to natural selection. We introduce a schematic, analytically solvable mathematical model to compare the adaptive value of different schemes of inheritance. Our model allows for variations to be inherited, randomly produced, or environmentally induced, and, irrespectively, to be either transmitted or not during reproduction. The adaptation of the different schemes for processing variations is quantified for a range of fluctuating environments, following an approach that links quantitative genetics with stochastic control theory.

Protein sectors: statistical coupling analysis versus conservation.

Statistical coupling analysis (SCA) is a method for analyzing multiple sequence alignments that was used to identify groups of coevolving residues termed "sectors". The method applies spectral analysis to a matrix obtained by combining correlation information with sequence conservation. It has been asserted that the protein sectors identified by SCA are functionally significant, with different sectors controlling different biochemical properties of the protein. Here we reconsider the available experimental data and note that it involves almost exclusively proteins with a single sector. We show that in this case sequence conservation is the dominating factor in SCA, and can alone be used to make statistically equivalent functional predictions. Therefore, we suggest shifting the experimental focus to proteins for which SCA identifies several sectors. Correlations in protein alignments, which have been shown to be informative in a number of independent studies, would then be less dominated by sequence conservation.

Behavioral diversity in microbes and low-dimensional phenotypic spaces.

Systematic studies of phenotypic diversity--required for understanding evolution--lag behind investigations of genetic diversity. Here we develop a quantitative approach to studying behavioral diversity, which we apply to swimming of the ciliate Tetrahymena. We measure the full-lifetime behavior of hundreds of individual organisms at high temporal resolution, over several generations and in diverse nutrient conditions. To characterize population diversity and temporal variability we introduce a unique statistical framework grounded in the notion of a phenotypic space of behaviors. We show that this space is effectively low dimensional with dimensions that correlate with a two-state "roaming and dwelling" model of swimming behavior. Temporal variability over the lifetime of an individual is correlated with the fraction of time spent roaming whereas diversity between individuals is correlated with the speed of roaming. Quantifying the dynamics of behavioral variation shows that behavior over the lifetime of an individual is strongly nonstationary. Analysis of behavioral dynamics between generations reveals complex patterns of behavioral heritability that point to the importance of considering correlations beyond mothers and daughters. Our description of a low-dimensional behavioral space should enable the systematic study of the evolutionary and ecological bases of phenotypic constraints. Future experimental and theoretical studies of behavioral diversity will have to account for the possibility of nonstationary and environmentally dependent behavioral dynamics that we observe.

Speed, dissipation, and error in kinetic proofreading.

Proofreading mechanisms increase specificity in biochemical reactions by allowing for the dissociation of intermediate complexes. These mechanisms disrupt and reset the reaction to undo errors at the cost of increased time of reaction and free energy expenditure. Here, we draw an analogy between proofreading and microtubule growth which share some of the features described above. Our analogy relates the statistics of growth and shrinkage of microtubules in physical space to the cycling of intermediate complexes in the space of chemical states in proofreading mechanisms. Using this analogy, we find a new kinetic regime of proofreading in which an exponential speed-up of the process can be achieved at the cost of a somewhat larger error rate. This regime is analogous to the transition region between two known growth regimes of microtubules (bounded and unbounded) and is sharply defined in the limit of large proofreading networks. We find that this advantageous regime of speed-error tradeoff might be present in proofreading schemes studied earlier in the charging of tRNA by tRNA synthetases, in RecA filament assembly on ssDNA, and in protein synthesis by ribosomes.

Individual histories and selection in heterogeneous populations.

The strength of selection in populations has traditionally been inferred by measuring changes in bulk population parameters, such as mean reproductive rates. Untangling the effect of selection from other factors, such as specific responses to environmental fluctuations, poses a significant problem both in microbiology and in other fields, including cancer biology and immunology, where selection occurs within phenotypically heterogeneous populations of cells. Using "individual histories"--temporal sequences of all reproduction events and phenotypic changes of individuals and their ancestors--we present an alternative approach to quantifying selection in diverse experimental settings. Selection is viewed as a process that acts on histories, and a measure of selection that employs the distribution of histories is introduced. We apply this measure to phenotypically structured populations in fluctuating environments across different evolutionary regimes. Additionally, we show that reproduction events alone, recorded in the population's tree of cell divisions, may be sufficient to accurately measure selection. The measure is thus applicable in a wide range of biological systems, from microorganisms--including species for which genetic tools do not yet exist--to cellular populations, such as tumors and stem cells, where detailed temporal data are becoming available.

Neuronal couplings between retinal ganglion cells inferred by efficient inverse statistical physics methods.

Complexity of neural systems often makes impracticable explicit measurements of all interactions between their constituents. Inverse statistical physics approaches, which infer effective couplings between neurons from their spiking activity, have been so far hindered by their computational complexity. Here, we present 2 complementary, computationally efficient inverse algorithms based on the Ising and "leaky integrate-and-fire" models. We apply those algorithms to reanalyze multielectrode recordings in the salamander retina in darkness and under random visual stimulus. We find strong positive couplings between nearby ganglion cells common to both stimuli, whereas long-range couplings appear under random stimulus only. The uncertainty on the inferred couplings due to limitations in the recordings (duration, small area covered on the retina) is discussed. Our methods will allow real-time evaluation of couplings for large assemblies of neurons.