Sloppiness, robustness, and evolvability in systems biology

A vast space of compact strategies for effective decisions

Inference-based decision-making, which underlies a broad range of behavioral tasks, is typically studied using a small number of handcrafted models. We instead enumerate a complete ensemble of strategies that could be used to effectively, but not necessarily optimally, solve a dynamic foraging task. Each strategy is expressed as a behavioral "program" that uses a limited number of internal states to specify actions conditioned on past observations. We show that the ensemble of strategies is enormous-comprising a quarter million programs with up to five internal states-but can nevertheless be understood in terms of algorithmic "mutations" that alter the structure of individual programs. We devise embedding algorithms that reveal how mutations away from a Bayesian-like strategy can diversify behavior while preserving performance, and we construct a compositional description to link low-dimensional changes in algorithmic structure with high-dimensional changes in behavior. Together, this work provides an alternative approach for understanding individual variability in behavior across animals and tasks.

Redundancy and the role of protein copy numbers in the cell polarization machinery of budding yeast

How can a self-organized cellular function evolve, adapt to perturbations, and acquire new sub-functions? To make progress in answering these basic questions of evolutionary cell biology, we analyze, as a concrete example, the cell polarity machinery of Saccharomyces cerevisiae. This cellular module exhibits an intriguing resilience: it remains operational under genetic perturbations and recovers quickly and reproducibly from the deletion of one of its key components. Using a combination of modeling, conceptual theory, and experiments, we propose that multiple, redundant self-organization mechanisms coexist within the protein network underlying cell polarization and are responsible for the module's resilience and adaptability. Based on our mechanistic understanding of polarity establishment, we hypothesize that scaffold proteins, by introducing new connections in the existing network, can increase the redundancy of mechanisms and thus increase the evolvability of other network components. Moreover, our work gives a perspective on how a complex, redundant cellular module might have evolved from a more rudimental ancestral form.

Heterogeneous off-target impact of ion-channel deletion on intrinsic properties of hippocampal model neurons that self-regulate calcium

How do neurons that implement cell-autonomous self-regulation of calcium react to knockout of individual ion-channel conductances? To address this question, we used a heterogeneous population of 78 conductance-based models of hippocampal pyramidal neurons that maintained cell-autonomous calcium homeostasis while receiving theta-frequency inputs. At calcium steady-state, we individually deleted each of the 11 active ion-channel conductances from each model. We measured the acute impact of deleting each conductance (one at a time) by comparing intrinsic electrophysiological properties before and immediately after channel deletion. The acute impact of deleting individual conductances on physiological properties (including calcium homeostasis) was heterogeneous, depending on the property, the specific model, and the deleted channel. The underlying many-to-many mapping between ion channels and properties pointed to ion-channel degeneracy. Next, we allowed the other conductances (barring the deleted conductance) to evolve towards achieving calcium homeostasis during theta-frequency activity. When calcium homeostasis was perturbed by ion-channel deletion, post-knockout plasticity in other conductances ensured resilience of calcium homeostasis to ion-channel deletion. These results demonstrate degeneracy in calcium homeostasis, as calcium homeostasis in knockout models was implemented in the absence of a channel that was earlier involved in the homeostatic process. Importantly, in reacquiring homeostasis, ion-channel conductances and physiological properties underwent heterogenous plasticity (dependent on the model, the property, and the deleted channel), even introducing changes in properties that were not directly connected to the deleted channel. Together, post-knockout plasticity geared towards maintaining homeostasis introduced heterogenous off-target effects on several channels and properties, suggesting that extreme caution be exercised in interpreting experimental outcomes involving channel knockouts.

Kinetic Modeling and Parameter Estimation of a Prebiotic Peptide Reaction Network

Although our understanding of how life emerged on Earth from simple organic precursors is speculative, early precursors likely included amino acids. The polymerization of amino acids into peptides and interactions between peptides are of interest because peptides and proteins participate in complex interaction networks in extant biology. However, peptide reaction networks can be challenging to study because of the potential for multiple species and systems-level interactions between species. We developed and employed a computational network model to describe reactions between amino acids to form di-, tri-, and tetra-peptides. Our experiments were initiated with two of the simplest amino acids, glycine and alanine, mediated by trimetaphosphate-activation and drying to promote peptide bond formation. The parameter estimates for bond formation and hydrolysis reactions in the system were found to be poorly constrained due to a network property known as sloppiness. In a sloppy model, the behavior mostly depends on only a subset of parameter combinations, but there is no straightforward way to determine which parameters should be included or excluded. Despite our inability to determine the exact values of specific kinetic parameters, we could make reasonably accurate predictions of model behavior. In short, our modeling has highlighted challenges and opportunities toward understanding the behaviors of complex prebiotic chemical experiments.

Predictive power of non-identifiable models

Resolving practical non-identifiability of computational models typically requires either additional data or non-algorithmic model reduction, which frequently results in models containing parameters lacking direct interpretation. Here, instead of reducing models, we explore an alternative, Bayesian approach, and quantify the predictive power of non-identifiable models. We considered an example biochemical signalling cascade model as well as its mechanical analogue. For these models, we demonstrated that by measuring a single variable in response to a properly chosen stimulation protocol, the dimensionality of the parameter space is reduced, which allows for predicting the measured variable's trajectory in response to different stimulation protocols even if all model parameters remain unidentified. Moreover, one can predict how such a trajectory will transform in the case of a multiplicative change of an arbitrary model parameter. Successive measurements of remaining variables further reduce the dimensionality of the parameter space and enable new predictions. We analysed potential pitfalls of the proposed approach that can arise when the investigated model is oversimplified, incorrect, or when the training protocol is inadequate. The main advantage of the suggested iterative approach is that the predictive power of the model can be assessed and practically utilised at each step.

Reproducibility and FAIR principles: the case of a segment polarity network model

The issue of reproducibility of computational models and the related FAIR principles (findable, accessible, interoperable, and reusable) are examined in a specific test case. I analyze a computational model of the segment polarity network in Drosophila embryos published in 2000. Despite the high number of citations to this publication, 23 years later the model is barely accessible, and consequently not interoperable. Following the text of the original publication allowed successfully encoding the model for the open source software COPASI. Subsequently saving the model in the SBML format allowed it to be reused in other open source software packages. Submission of this SBML encoding of the model to the BioModels database enables its findability and accessibility. This demonstrates how the FAIR principles can be successfully enabled by using open source software, widely adopted standards, and public repositories, facilitating reproducibility and reuse of computational cell biology models that will outlive the specific software used.

Reliable and robust control of nucleus centering is contingent on nonequilibrium force patterns

Cell centers their division apparatus to ensure symmetric cell division, a challenging task when the governing dynamics is stochastic. Using fission yeast, we show that the patterning of nonequilibrium polymerization forces of microtubule (MT) bundles controls the precise localization of spindle pole body (SPB), and hence the division septum, at the onset of mitosis. We define two cellular objectives, reliability, the mean SPB position relative to the geometric center, and robustness, the variance of the SPB position, which are sensitive to genetic perturbations that change cell length, MT bundle number/orientation, and MT dynamics. We show that simultaneous control of reliability and robustness is required to minimize septum positioning error achieved by the wild type (WT). A stochastic model for the MT-based nucleus centering, with parameters measured directly or estimated using Bayesian inference, recapitulates the maximum fidelity of WT. Using this, we perform a sensitivity analysis of the parameters that control nuclear centering.

Far from Asymptopia: Unbiased High-Dimensional Inference Cannot Assume Unlimited Data

Inference from limited data requires a notion of measure on parameter space, which is most explicit in the Bayesian framework as a prior distribution. Jeffreys prior is the best-known uninformative choice, the invariant volume element from information geometry, but we demonstrate here that this leads to enormous bias in typical high-dimensional models. This is because models found in science typically have an effective dimensionality of accessible behaviors much smaller than the number of microscopic parameters. Any measure which treats all of these parameters equally is far from uniform when projected onto the sub-space of relevant parameters, due to variations in the local co-volume of irrelevant directions. We present results on a principled choice of measure which avoids this issue and leads to unbiased posteriors by focusing on relevant parameters. This optimal prior depends on the quantity of data to be gathered, and approaches Jeffreys prior in the asymptotic limit. However, for typical models, this limit cannot be justified without an impossibly large increase in the quantity of data, exponential in the number of microscopic parameters.

Sloppiness: Fundamental study, new formalism and its application in model assessment

Computational modelling of biological processes poses multiple challenges in each stage of the modelling exercise. Some significant challenges include identifiability, precisely estimating parameters from limited data, informative experiments and anisotropic sensitivity in the parameter space. One of these challenges' crucial but inconspicuous sources is the possible presence of large regions in the parameter space over which model predictions are nearly identical. This property, known as sloppiness, has been reasonably well-addressed in the past decade, studying its possible impacts and remedies. However, certain critical unanswered questions concerning sloppiness, particularly related to its quantification and practical implications in various stages of system identification, still prevail. In this work, we systematically examine sloppiness at a fundamental level and formalise two new theoretical definitions of sloppiness. Using the proposed definitions, we establish a mathematical relationship between the parameter estimates' precision and sloppiness in linear predictors. Further, we develop a novel computational method and a visual tool to assess the goodness of a model around a point in parameter space by identifying local structural identifiability and sloppiness and finding the most sensitive and least sensitive parameters for non-infinitesimal perturbations. We demonstrate the working of our method in benchmark systems biology models of various complexities. The pharmacokinetic HIV infection model analysis identified a new set of biologically relevant parameters that can be used to control the free virus in an active HIV infection.

Information geometry for multiparameter models: new perspectives on the origin of simplicity

Complex models in physics, biology, economics, and engineering are oftensloppy, meaning that the model parameters are not well determined by the model predictions for collective behavior. Many parameter combinations can vary over decades without significant changes in the predictions. This review uses information geometry to explore sloppiness and its deep relation to emergent theories. We introduce themodel manifoldof predictions, whose coordinates are the model parameters. Itshyperribbonstructure explains why only a few parameter combinations matter for the behavior. We review recent rigorous results that connect the hierarchy of hyperribbon widths to approximation theory, and to the smoothness of model predictions under changes of the control variables. We discuss recent geodesic methods to find simpler models on nearby boundaries of the model manifold-emergent theories with fewer parameters that explain the behavior equally well. We discuss a Bayesian prior which optimizes the mutual information between model parameters and experimental data, naturally favoring points on the emergent boundary theories and thus simpler models. We introduce a 'projected maximum likelihood' prior that efficiently approximates this optimal prior, and contrast both to the poor behavior of the traditional Jeffreys prior. We discuss the way the renormalization group coarse-graining in statistical mechanics introduces a flow of the model manifold, and connect stiff and sloppy directions along the model manifold with relevant and irrelevant eigendirections of the renormalization group. Finally, we discuss recently developed 'intensive' embedding methods, allowing one to visualize the predictions of arbitrary probabilistic models as low-dimensional projections of an isometric embedding, and illustrate our method by generating the model manifold of the Ising model.

Open problems in mathematical biology

Biology is data-rich, and it is equally rich in concepts and hypotheses. Part of trying to understand biological processes and systems is therefore to confront our ideas and hypotheses with data using statistical methods to determine the extent to which our hypotheses agree with reality. But doing so in a systematic way is becoming increasingly challenging as our hypotheses become more detailed, and our data becomes more complex. Mathematical methods are therefore gaining in importance across the life- and biomedical sciences. Mathematical models allow us to test our understanding, make testable predictions about future behaviour, and gain insights into how we can control the behaviour of biological systems. It has been argued that mathematical methods can be of great benefit to biologists to make sense of data. But mathematics and mathematicians are set to benefit equally from considering the often bewildering complexity inherent to living systems. Here we present a small selection of open problems and challenges in mathematical biology. We have chosen these open problems because they are of both biological and mathematical interest.

A data-assimilation approach to predict population dynamics during epithelial-mesenchymal transition

Epithelial-mesenchymal transition (EMT) is a biological process that plays a central role in embryonic development, tissue regeneration, and cancer metastasis. Transforming growth factor-β (TGFβ) is a potent inducer of this cellular transition, comprising transitions from an epithelial state to partial or hybrid EMT state(s), to a mesenchymal state. Recent experimental studies have shown that, within a population of epithelial cells, heterogeneous phenotypical profiles arise in response to different time- and TGFβ dose-dependent stimuli. This offers a challenge for computational models, as most model parameters are generally obtained to represent typical cell responses, not necessarily specific responses nor to capture population variability. In this study, we applied a data-assimilation approach that combines limited noisy observations with predictions from a computational model, paired with parameter estimation. Synthetic experiments mimic the biological heterogeneity in cell states that is observed in epithelial cell populations by generating a large population of model parameter sets. Analysis of the parameters for virtual epithelial cells with biologically significant characteristics (e.g., EMT prone or resistant) illustrates that these sub-populations have identifiable critical model parameters. We perform a series of in silico experiments in which a forecasting system reconstructs the EMT dynamics of each virtual cell within a heterogeneous population exposed to time-dependent exogenous TGFβ dose and either an EMT-suppressing or EMT-promoting perturbation. We find that estimating population-specific critical parameters significantly improved the prediction accuracy of cell responses. Thus, with appropriate protocol design, we demonstrate that a data-assimilation approach successfully reconstructs and predicts the dynamics of a heterogeneous virtual epithelial cell population in the presence of physiological model error and parameter uncertainty.

Signal neutrality, scalar property, and collapsing boundaries as consequences of a learned multi-timescale strategy

We postulate that three fundamental elements underlie a decision making process: perception of time passing, information processing in multiple timescales and reward maximisation. We build a simple reinforcement learning agent upon these principles that we train on a random dot-like task. Our results, similar to the experimental data, demonstrate three emerging signatures. (1) signal neutrality: insensitivity to the signal coherence in the interval preceding the decision. (2) Scalar property: the mean of the response times varies widely for different signal coherences, yet the shape of the distributions stays almost unchanged. (3) Collapsing boundaries: the “effective” decision-making boundary changes over time in a manner reminiscent of the theoretical optimal. Removing the perception of time or the multiple timescales from the model does not preserve the distinguishing signatures. Our results suggest an alternative explanation for signal neutrality. We propose that it is not part of motor planning. It is part of the decision-making process and emerges from information processing on multiple timescales.

The sloppy relationship between neural circuit structure and function

Investigating and describing the relationships between the structure of a circuit and its function has a long tradition in neuroscience. Since neural circuits acquire their structure through sophisticated developmental programmes, and memories and experiences are maintained through synaptic modification, it is to be expected that structure is closely linked to function. Recent findings challenge this hypothesis from three different angles: Function does not strongly constrain circuit parameters, many parameters in neural circuits are irrelevant and contribute little to function, and circuit parameters are unstable and subject to constant random drift. At the same time however, recent work also showed that dynamics in neural circuit activity that is related to function are robust over time and across individuals. Here this apparent contradiction is addressed by considering the properties of neural manifolds that restrict circuit activity to functionally relevant subspaces, and it will be suggested that degenerate, anisotropic and unstable parameter spaces are a closely related to the structure and implementation of functionally relevant neural manifolds. Abstract figure legend What are the relationships between noisy and highly variable microscopic neural circuit variables on the one hand and the generation of behaviour on the other? Here it is proposed that an intermediate level of description exists where this relationship can be understood in terms of low-dimensional dynamics. Recordings of neural activity during unconstrained behaviour and the development of new machine learning methods will help to uncover these links. This article is protected by copyright. All rights reserved.

Application of Sensitivity Analysis to Discover Potential Molecular Drug Targets

Mathematical modeling of signaling pathways and regulatory networks has been supporting experimental research for some time now. Sensitivity analysis, aimed at finding model parameters whose changes yield significantly altered cellular responses, is an important part of modeling work. However, sensitivity methods are often directly transplanted from analysis of technical systems, and thus, they may not serve the purposes of analysis of biological systems. This paper presents a novel sensitivity analysis method that is particularly suited to the task of searching for potential molecular drug targets in signaling pathways. Using two sample models of pathways, p53/Mdm2 regulatory module and IFN-β-induced JAK/STAT signaling pathway, we show that the method leads to biologically relevant conclusions, identifying processes suitable for targeted pharmacological inhibition, represented by the reduction of kinetic parameter values. That, in turn, facilitates subsequent search for active drug components.

Are some species ‘robust’ to exploitation? Explaining persistence in deceptive relationships

Animals and plants trick others in an extraordinary diversity of ways to gain fitness benefits. Mimicry and deception can, for example, lure prey, reduce the costs of parental care or aid in pollination–in ways that impose fitness costs on the exploited party. The evolutionary maintenance of such asymmetric relationships often relies on these costs being mitigated through counter-adaptations, low encounter rates, or indirect fitness benefits. However, these mechanisms do not always explain the evolutionary persistence of some classic deceptive interactions.

Sexually deceptive pollination (in which plants trick male pollinators into mating with their flowers) has evolved multiple times independently, mainly in the southern hemisphere and especially in Australasia and Central and South America. This trickery imposes considerable costs on the males: they miss out on mating opportunities, and in some cases, waste their limited sperm on the flower. These relationships appear stable, yet in some cases there is little evidence suggesting that their persistence relies on counter-adaptations, low encounter rates, or indirect fitness benefits. So, how might these relationships persist?

Here, we introduce and explore an additional hypothesis from systems biology: that some species are robust to exploitation. Robustness arises from a species’ innate traits and means they are robust against costs of exploitation. This allows species to persist where a population without those traits would not, making them ideal candidates for exploitation. We propose that this mechanism may help inform new research approaches and provide insight into how exploited species might persist.

Multiple timescales in bacterial growth homeostasis

In balanced exponential growth, bacteria maintain many properties statistically stable for a long time: cell size, cell cycle time, and more. As these are strongly coupled variables, it is not a-priori obvious which are directly regulated and which are stabilized through interactions. Here, we address this problem by separating timescales in bacterial single-cell dynamics. Disentangling homeostatic set points from fluctuations around them reveals that some variables, such as growth-rate, cell size and cycle time, are "sloppy" with highly volatile set points. Quantifying the relative contribution of environmental and internal sources, we find that sloppiness is primarily driven by the environment. Other variables such as fold-change define "stiff" combinations of coupled variables with robust set points. These results are manifested geometrically as a control manifold in the space of variables: set points span a wide range of values within the manifold, whereas out-of-manifold deviations are constrained. Our work offers a generalizable data-driven approach for identifying control variables in a multidimensional system.

Petri nets and ODEs as complementary methods for comprehensive analysis on an example of the ATM-p53-NF-[Formula: see text]B signaling pathways

Intracellular processes are cascades of biochemical reactions, triggered in response to various types of stimuli. Mathematical models describing their dynamics have become increasingly popular in recent years, as tools supporting experimental work in analysis of pathways and regulatory networks. Not only do they provide insights into general properties of these systems, but also help in specific tasks, such as search for drug molecular targets or treatment protocols. Different tools and methods are used to model complex biological systems. In this work, we focus on ordinary differential equations (ODEs) and Petri nets. We consider specific methods of analysis of such models, i.e., sensitivity analysis (SA) and significance analysis. So far, they have been applied separately, with different goals. In this paper, we show that they can complement each other, combining the sensitivity of ODE models and the significance analysis of Petri nets. The former is used to find parameters, whose change results in the greatest quantitative and qualitative changes in the model response, while the latter is a structural analysis and allows indicating the most important subprocesses in terms of information flow in Petri net. Ultimately, both methods facilitate finding the essential processes in a given signaling pathway or regulatory network and may be used to support medical therapy development. In the paper, the use of dual modeling is illustrated with an example of ATM/p53/NF-[Formula: see text]B pathway. Each method was applied to analyze this system, resulting in finding different subsets of important processes that might be prospective targets for changing this system behavior. While some of the processes were indicated in each of the approaches, others were found by one method only and would be missed if only that method was applied. This leads to the conclusion about the complementarity of the methods under investigation. The dual modeling approach of comprehensive structural and parametric analysis yields results that would not be possible if these two modeling approaches were applied separately. The combined approach, proposed in this paper, facilitates finding not only key processes, with which significant parameters are associated, but also significant modules, corresponding to subsystems of regulatory networks. The results provide broader insight into therapy targets in diseases in which the natural control of intracellular processes is disturbed, leading to the development of more effective therapies in medicine.

The information theory of developmental pruning: Optimizing global network architectures using local synaptic rules

During development, biological neural networks produce more synapses and neurons than needed. Many of these synapses and neurons are later removed in a process known as neural pruning. Why networks should initially be over-populated, and the processes that determine which synapses and neurons are ultimately pruned, remains unclear. We study the mechanisms and significance of neural pruning in model neural networks. In a deep Boltzmann machine model of sensory encoding, we find that (1) synaptic pruning is necessary to learn efficient network architectures that retain computationally-relevant connections, (2) pruning by synaptic weight alone does not optimize network size and (3) pruning based on a locally-available measure of importance based on Fisher information allows the network to identify structurally important vs. unimportant connections and neurons. This locally-available measure of importance has a biological interpretation in terms of the correlations between presynaptic and postsynaptic neurons, and implies an efficient activity-driven pruning rule. Overall, we show how local activity-dependent synaptic pruning can solve the global problem of optimizing a network architecture. We relate these findings to biology as follows: (I) Synaptic over-production is necessary for activity-dependent connectivity optimization. (II) In networks that have more neurons than needed, cells compete for activity, and only the most important and selective neurons are retained. (III) Cells may also be pruned due to a loss of synapses on their axons. This occurs when the information they convey is not relevant to the target population.

Biological neural networks need to be efficient and compact, as synapses and neurons require space to store and energy to operate and maintain. This favors an optimized network topology that minimizes redundant neurons and connections. Large numbers of extra neurons and synapses are produced during development, and later removed as the brain matures. A key question to understand this process is how neurons determine which synapses are important. We used statistical models of neural networks to simulate developmental pruning. We show that neurons in such networks can use locally available information to measure the importance of their synapses in a biologically plausible way. We demonstrate that this pruning rule, which is motivated by information theoretic considerations, retains network topologies that can efficiently encode sensory inputs. In contrast, pruning at random, or based on synaptic weights alone, was less able to identify redundant neurons.

Programmatic modeling for biological systems

Computational modeling has become an established technique to encode mathematical representations of cellular processes and gain mechanistic insights that drive testable predictions. These models are often constructed using graphical user interfaces or domain-specific languages, with community standards used for interchange. Models undergo steady state or dynamic analysis, which can include simulation and calibration within a single application, or transfer across various tools. Here, we describe a novel programmatic modeling paradigm, whereby modeling is augmented with software engineering best practices. We focus on Python - a popular programming language with a large scientific package ecosystem. Models can be encoded as programs, adding benefits such as modularity, testing, and automated documentation generators, while still being extensible and exportable to standardized formats for use with external tools if desired. Programmatic modeling is a key technology to enable collaborative model development and enhance dissemination, transparency, and reproducibility.

Dynamics-Evolution Correspondence in Protein Structures

The genotype-phenotype mapping of proteins is a fundamental question in structural biology. In this Letter, with the analysis of a large dataset of proteins from hundreds of protein families, we quantitatively demonstrate the correlations between the noise-induced protein dynamics and mutation-induced variations of native structures, indicating the dynamics-evolution correspondence of proteins. Based on the investigations of the linear responses of native proteins, the origin of such a correspondence is elucidated. It is essential that the noise- and mutation-induced deformations of the proteins are restricted on a common low-dimensional subspace, as confirmed from the data. These results suggest an evolutionary mechanism of the proteins gaining both dynamical flexibility and evolutionary structural variability.

Parameter estimation in fluorescence recovery after photobleaching: quantitative analysis of protein binding reactions and diffusion

Fluorescence recovery after photobleaching (FRAP) is a common experimental method for investigating rates of molecular redistribution in biological systems. Many mathematical models of FRAP have been developed, the purpose of which is usually the estimation of certain biological parameters such as the diffusivity and chemical reaction rates of a protein, this being accomplished by fitting the model to experimental data. In this article, we consider a two species reaction–diffusion FRAP model. Using asymptotic analysis, we derive new FRAP recovery curve approximation formulae, and formally re-derive existing ones. On the basis of these formulae, invoking the concept of Fisher information, we predict, in terms of biological and experimental parameters, sufficient conditions to ensure that the values all model parameters can be estimated from data. We verify our predictions with extensive computational simulations. We also use computational methods to investigate cases in which some or all biological parameters are theoretically inestimable. In these cases, we propose methods which can be used to extract the maximum possible amount of information from the FRAP data.

Constraint-based metabolic control analysis for rational strain engineering

The advancements in genome editing techniques over the past years have rekindled interest in rational metabolic engineering strategies. While Metabolic Control Analysis (MCA) is a well-established method for quantifying the effects of metabolic engineering interventions on flows in metabolic networks and metabolite concentrations, it does not consider the physiological limitations of the cellular environment and metabolic engineering design constraints. We report here a constraint-based framework, Network Response Analysis (NRA), for rational genetic strain design. NRA is cast as a Mixed-Integer Linear Programming problem that integrates MCA, Thermodynamically-based Flux Analysis (TFA), biologically relevant constraints, as well as genome editing restrictions into a comprehensive platform for identifying metabolic engineering targets. We show that the NRA formulation and its core constraints are equivalent to the ones of Flux Balance Analysis (FBA) and TFA, which allows it to be used for a wide range of optimization criteria and with various physiological constraints. We also show how the parametrization and introduction of biological constraints enhance the NRA formulation compared to the classical MCA approach, and we demonstrate its features and its ability to generate multiple alternative optimal strategies given several user-defined boundaries and objectives. In summary, NRA is a sophisticated alternative to classical MCA for rational metabolic engineering that accommodates the incorporation of physiological data at metabolic flux, metabolite concentration, and enzyme expression levels.

A Life Cycle for Modeling Biology at Different Scales

Modeling has become a popular tool for inquiry and discovery across biological disciplines. Models allow biologists to probe complex questions and to guide experimentation. Modeling literacy among biologists, however, has not always kept pace with the rise in popularity of these techniques and the relevant advances in modeling theory. The result is a lack of understanding that inhibits communication and ultimately, progress in data gathering and analysis. In an effort to help bridge this gap, we present a blueprint that will empower biologists to interrogate and apply models in their field. We demonstrate the applicability of this blueprint in two case studies from distinct subdisciplines of biology; developmental-biomechanics and evolutionary biology. The models used in these fields vary from summarizing dynamical mechanisms to making statistical inferences, demonstrating the breadth of the utility of models to explore biological phenomena.

Optimal Encoding in Stochastic Latent-Variable Models

In this work we explore encoding strategies learned by statistical models of sensory coding in noisy spiking networks. Early stages of sensory communication in neural systems can be viewed as encoding channels in the information-theoretic sense. However, neural populations face constraints not commonly considered in communications theory. Using restricted Boltzmann machines as a model of sensory encoding, we find that networks with sufficient capacity learn to balance precision and noise-robustness in order to adaptively communicate stimuli with varying information content. Mirroring variability suppression observed in sensory systems, informative stimuli are encoded with high precision, at the cost of more variable responses to frequent, hence less informative stimuli. Curiously, we also find that statistical criticality in the neural population code emerges at model sizes where the input statistics are well captured. These phenomena have well-defined thermodynamic interpretations, and we discuss their connection to prevailing theories of coding and statistical criticality in neural populations.

Dynamic clamp constructed phase diagram for the Hodgkin and Huxley model of excitability

Excitability-a threshold-governed transient in transmembrane voltage-is a fundamental physiological process that controls the function of the heart, endocrine, muscles, and neuronal tissues. The 1950s Hodgkin and Huxley explicit formulation provides a mathematical framework for understanding excitability, as the consequence of the properties of voltage-gated sodium and potassium channels. The Hodgkin-Huxley model is more sensitive to parametric variations of protein densities and kinetics than biological systems whose excitability is apparently more robust. It is generally assumed that the model's sensitivity reflects missing functional relations between its parameters or other components present in biological systems. Here we experimentally assembled excitable membranes using the dynamic clamp and voltage-gated potassium ionic channels (Kv1.3) expressed in Xenopus oocytes. We take advantage of a theoretically derived phase diagram, where the phenomenon of excitability is reduced to two dimensions defined as combinations of the Hodgkin-Huxley model parameters, to examine functional relations in the parameter space. Moreover, we demonstrate activity dependence and hysteretic dynamics over the phase diagram due to the impacts of complex slow inactivation kinetics. The results suggest that maintenance of excitability amid parametric variation is a low-dimensional, physiologically tenable control process. In the context of model construction, the results point to a potentially significant gap between high-dimensional models that capture the full measure of complexity displayed by ion channel function and the lower dimensionality that captures physiological function.

Unwinding the model manifold: Choosing similarity measures to remove local minima in sloppy dynamical systems

In this paper, we consider the problem of parameter sensitivity in models of complex dynamical systems through the lens of information geometry. We calculate the sensitivity of model behavior to variations in parameters. In most cases, models are sloppy, that is, exhibit an exponential hierarchy of parameter sensitivities. We propose a parameter classification scheme based on how the sensitivities scale at long observation times. We show that for oscillatory models, either with a limit cycle or a strange attractor, sensitivities can become arbitrarily large, which implies a high effective dimensionality on the model manifold. Sloppy models with a single fixed point have model manifolds with low effective dimensionality, previously described as a "hyper-ribbon." In contrast, models with high effective dimensionality translate into multimodal fitting problems. We define a measure of curvature on the model manifold which we call the winding frequency that estimates the density of local minima in the model's parameter space. We then show how alternative choices of fitting metrics can "unwind" the model manifold and give low winding frequencies. This prescription translates the model manifold from one of high effective dimensionality into the hyper-ribbon structures observed elsewhere. This translation opens the door for applications of sloppy model analysis and model reduction methods developed for models with low effective dimensionality.

Adding levels of complexity enhances robustness and evolvability in a multilevel genotype-phenotype map

Robustness and evolvability are the main properties that account for the stability and accessibility of phenotypes. They have been studied in a number of computational genotype-phenotype maps. In this paper, we study a metabolic genotype-phenotype map defined in toyLIFE, a multilevel computational model that represents a simplified cellular biology. toyLIFE includes several levels of phenotypic expression, from proteins to regulatory networks to metabolism. Our results show that toyLIFE shares many similarities with other seemingly unrelated computational genotype-phenotype maps. Thus, toyLIFE shows a high degeneracy in the mapping from genotypes to phenotypes, as well as a highly skewed distribution of phenotypic abundances. The neutral networks associated with abundant phenotypes are highly navigable, and common phenotypes are close to each other in genotype space. All of these properties are remarkable, as toyLIFE is built on a version of the HP protein-folding model that is neither robust nor evolvable: phenotypes cannot be mutually accessed through point mutations. In addition, both robustness and evolvability increase with the number of genes in a genotype. Therefore, our results suggest that adding levels of complexity to the mapping of genotypes to phenotypes and increasing genome size enhances both these properties.

Parameter estimation and identifiability in a neural population model for electro-cortical activity

Electroencephalography (EEG) provides a non-invasive measure of brain electrical activity. Neural population models, where large numbers of interacting neurons are considered collectively as a macroscopic system, have long been used to understand features in EEG signals. By tuning dozens of input parameters describing the excitatory and inhibitory neuron populations, these models can reproduce prominent features of the EEG such as the alpha-rhythm. However, the inverse problem, of directly estimating the parameters from fits to EEG data, remains unsolved. Solving this multi-parameter non-linear fitting problem will potentially provide a real-time method for characterizing average neuronal properties in human subjects. Here we perform unbiased fits of a 22-parameter neural population model to EEG data from 82 individuals, using both particle swarm optimization and Markov chain Monte Carlo sampling. We estimate how much is learned about individual parameters by computing Kullback-Leibler divergences between posterior and prior distributions for each parameter. Results indicate that only a single parameter, that determining the dynamics of inhibitory synaptic activity, is directly identifiable, while other parameters have large, though correlated, uncertainties. We show that the eigenvalues of the Fisher information matrix are roughly uniformly spaced over a log scale, indicating that the model is sloppy, like many of the regulatory network models in systems biology. These eigenvalues indicate that the system can be modeled with a low effective dimensionality, with inhibitory synaptic activity being prominent in driving system behavior.

Landscape of gene networks for random parameter perturbation

Landscape approaches have been exploited to study the stochastic dynamics of gene networks. However, how to calculate the landscape with a wide range of parameter variations and how to investigate the influence of the network topology on the global properties of gene networks remain to be elucidated. Here, I developed an approach for the landscape of random parameter perturbation (LRPP) to address this issue. Based on a self-consistent approximation approach, by making perturbations to parameters in a given range, I obtained the landscape for gene network systems. I applied this approach to two biological models, one for the mutual repression model and the other for the embryonic stem (ES) cell differentiation network. For the mutual repression model, my results confirm quantitatively that positive feedback promotes the robustness of multistability. For the ES cell differentiation model, I identify three cell states, representing the ES cell, the differentiation cell, and the intermediate state cell, respectively. I propose that the intermediate states and the wide range of parameter values coming from inhomogeneous cellular environments provide possible explanations for the heterogeneity observed in single cell experiments. I also offer a counterintuitive result that noise could reduce heterogeneity and promote the stability of cell states. These results support that the network topology determines the operating principles of the genetic networks, reflected by the representative landscapes from LRPP. This work provides a new route to obtain the potential landscape for a gene network system given a wide range of parameter values and study the influences of the network topology on the global properties of the system.

Formation of dominant mode by evolution in biological systems

A reduction in high-dimensional phenotypic states to a few degrees of freedom is essential to understand biological systems. Here, we show evolutionary robustness causes such reduction which restricts possible phenotypic changes in response to a variety of environmental conditions. First, global protein expression changes in Escherichia coli after various environmental perturbations were shown to be proportional across components, across different types of environmental conditions. To examine if such dimension reduction is a result of evolution, we analyzed a cell model-with a huge number of components, that reproduces itself via a catalytic reaction network-and confirmed that common proportionality in the concentrations of all components is shaped through evolutionary processes. We found that the changes in concentration across all components in response to environmental and evolutionary changes are constrained to the changes along a one-dimensional major axis, within a huge-dimensional state space. On the basis of these observations, we propose a theory in which such constraints in phenotypic changes are achieved both by evolutionary robustness and plasticity and formulate this proposition in terms of dynamical systems. Accordingly, broad experimental and numerical results on phenotypic changes caused by evolution and adaptation are coherently explained.

The Ca2+ transient as a feedback sensor controlling cardiomyocyte ionic conductances in mouse populations

Conductances of ion channels and transporters controlling cardiac excitation may vary in a population of subjects with different cardiac gene expression patterns. However, the amount of variability and its origin are not quantitatively known. We propose a new conceptual approach to predict this variability that consists of finding combinations of conductances generating a normal intracellular Ca²⁺ transient without any constraint on the action potential. Furthermore, we validate experimentally its predictions using the Hybrid Mouse Diversity Panel, a model system of genetically diverse mouse strains that allows us to quantify inter-subject versus intra-subject variability. The method predicts that conductances of inward Ca²⁺ and outward K⁺ currents compensate each other to generate a normal Ca²⁺ transient in good quantitative agreement with current measurements in ventricular myocytes from hearts of different isogenic strains. Our results suggest that a feedback mechanism sensing the aggregate Ca²⁺ transient of the heart suffices to regulate ionic conductances.

Robustness of a Reconstituted Escherichia coli Protein Translation System Analyzed by Computational Modeling

Robustness against environmental changes is one of the major features of biological systems, but its origin is not well understood. We recently constructed a large-scale computational model of an Escherichia coli-based reconstituted in vitro translation system that enumerates all protein synthesis processes in detail. Our model synthesizes a formyl-Met-Gly-Gly tripeptide (MGG peptide) from 27 initial molecular components through 968 biochemical reactions. Among the 968 kinetic parameters, 483 are nonzero parameters, and the simulator was used to determine how perturbations of 483 individual reactions affect the complex reaction network. We found that even when the kinetic parameter was changed from 100- to 0.01-fold, 94% of the changes hardly affected the two indicators of reaction dynamics in MGG peptide synthesis, which represent the yield of the MGG peptide and the initial lag-time of the peptide synthesis. Moreover, none of the indicators increased proportionally to these changes: e.g., a 100-fold increase in the kinetic parameter increased the yield by only 2.2-fold at most, indicating the insensitivity of the reaction network to perturbation. Robustness and insensitivity are likely to be a common feature of large-scale biological reaction networks.

Individuality and slow dynamics in bacterial growth homeostasis

Microbial cells go through repeated cycles of growth and division. These cycles are not perfect: the time and size at division can fluctuate from one cycle to the next. Still, cell size is kept tightly controlled, and fluctuations do not accumulate to large deviations. How this control is implemented in single cells is still not fully understood. We performed experiments that follow individual bacteria in microfluidic traps for hundreds of generations. This enables us to identify distinct individual dynamic properties that are maintained over many cycles of growth and division. Surprisingly, we find that each cell suppresses fluctuations with a different strength; this variability defines an “individual” behavior for each cell, which is inherited along many generations.

Microbial growth and division are fundamental processes relevant to many areas of life science. Of particular interest are homeostasis mechanisms, which buffer growth and division from accumulating fluctuations over multiple cycles. These mechanisms operate within single cells, possibly extending over several division cycles. However, all experimental studies to date have relied on measurements pooled from many distinct cells. Here, we disentangle long-term measured traces of individual cells from one another, revealing subtle differences between temporal and pooled statistics. By analyzing correlations along up to hundreds of generations, we find that the parameter describing effective cell size homeostasis strength varies significantly among cells. At the same time, we find an invariant cell size, which acts as an attractor to all individual traces, albeit with different effective attractive forces. Despite the common attractor, each cell maintains a distinct average size over its finite lifetime with suppressed temporal fluctuations around it, and equilibration to the global average size is surprisingly slow (>150 cell cycles). To show a possible source of variable homeostasis strength, we construct a mathematical model relying on intracellular interactions, which integrates measured properties of cell size with those of highly expressed proteins. Effective homeostasis strength is then influenced by interactions and by noise levels and generally varies among cells. A predictable and measurable consequence of variable homeostasis strength appears as distinct oscillatory patterns in cell size and protein content over many generations. We discuss implications of our results to understanding mechanisms controlling division in single cells and their characteristic timescales.

Macroscopic Theory for Evolving Biological Systems Akin to Thermodynamics

We present a macroscopic theory to characterize the plasticity, robustness, and evolvability of biological responses and their fluctuations. First, linear approximation in intracellular reaction dynamics is used to demonstrate proportional changes in the expression of all cellular components in response to a given environmental stress, with the proportion coefficient determined by the change in growth rate as a consequence of the steady growth of cells. We further demonstrate that this relationship is supported through adaptation experiments of bacteria, perhaps too well as this proportionality is held even across cultures of different types of conditions. On the basis of simulations of cell models, we further show that this global proportionality is a consequence of evolution in which expression changes in response to environmental or genetic perturbations are constrained along a unique one-dimensional curve, which is a result of evolutionary robustness. It then follows that the expression changes induced by environmental changes are proportionally reduced across different components of a cell by evolution, which is akin to the Le Chatelier thermodynamics principle. Finally, with the aid of a fluctuation-response relationship, this proportionality is shown to hold between fluctuations caused by genetic changes and those caused by noise. Overall, these results and support from the theoretical and experimental literature suggest a formulation of cellular systems akin to thermodynamics, in which a macroscopic potential is given by the growth rate (or fitness) represented as a function of environmental and evolutionary changes.

Identification of gene regulation models from single-cell data

In quantitative analyses of biological processes, one may use many different scales of models (e.g. spatial or non-spatial, deterministic or stochastic, time-varying or at steady-state) or many different approaches to match models to experimental data (e.g. model fitting or parameter uncertainty/sloppiness quantification with different experiment designs). These different analyses can lead to surprisingly different results, even when applied to the same data and the same model. We use a simplified gene regulation model to illustrate many of these concerns, especially for ODE analyses of deterministic processes, chemical master equation and finite state projection analyses of heterogeneous processes, and stochastic simulations. For each analysis, we employ Matlab and Python software to consider a time-dependent input signal (e.g. a kinase nuclear translocation) and several model hypotheses, along with simulated single-cell data. We illustrate different approaches (e.g. deterministic and stochastic) to identify the mechanisms and parameters of the same model from the same simulated data. For each approach, we explore how uncertainty in parameter space varies with respect to the chosen analysis approach or specific experiment design. We conclude with a discussion of how our simulated results relate to the integration of experimental and computational investigations to explore signal-activated gene expression models in yeast (Neuert et al 2013 Science 339 584-7) and human cells (Senecal et al 2014 Cell Rep. 8 75-83)5.

Simulating biological processes: stochastic physics from whole cells to colonies

The last few decades have revealed the living cell to be a crowded spatially heterogeneous space teeming with biomolecules whose concentrations and activities are governed by intrinsically random forces. It is from this randomness, however, that a vast array of precisely timed and intricately coordinated biological functions emerge that give rise to the complex forms and behaviors we see in the biosphere around us. This seemingly paradoxical nature of life has drawn the interest of an increasing number of physicists, and recent years have seen stochastic modeling grow into a major subdiscipline within biological physics. Here we review some of the major advances that have shaped our understanding of stochasticity in biology. We begin with some historical context, outlining a string of important experimental results that motivated the development of stochastic modeling. We then embark upon a fairly rigorous treatment of the simulation methods that are currently available for the treatment of stochastic biological models, with an eye toward comparing and contrasting their realms of applicability, and the care that must be taken when parameterizing them. Following that, we describe how stochasticity impacts several key biological functions, including transcription, translation, ribosome biogenesis, chromosome replication, and metabolism, before considering how the functions may be coupled into a comprehensive model of a 'minimal cell'. Finally, we close with our expectation for the future of the field, focusing on how mesoscopic stochastic methods may be augmented with atomic-scale molecular modeling approaches in order to understand life across a range of length and time scales.

Epigenetic regulation of cell fate reprogramming in aging and disease: A predictive computational model

Understanding the control of epigenetic regulation is key to explain and modify the aging process. Because histone-modifying enzymes are sensitive to shifts in availability of cofactors (e.g. metabolites), cellular epigenetic states may be tied to changing conditions associated with cofactor variability. The aim of this study is to analyse the relationships between cofactor fluctuations, epigenetic landscapes, and cell state transitions. Using Approximate Bayesian Computation, we generate an ensemble of epigenetic regulation (ER) systems whose heterogeneity reflects variability in cofactor pools used by histone modifiers. The heterogeneity of epigenetic metabolites, which operates as regulator of the kinetic parameters promoting/preventing histone modifications, stochastically drives phenotypic variability. The ensemble of ER configurations reveals the occurrence of distinct epi-states within the ensemble. Whereas resilient states maintain large epigenetic barriers refractory to reprogramming cellular identity, plastic states lower these barriers, and increase the sensitivity to reprogramming. Moreover, fine-tuning of cofactor levels redirects plastic epigenetic states to re-enter epigenetic resilience, and vice versa. Our ensemble model agrees with a model of metabolism-responsive loss of epigenetic resilience as a cellular aging mechanism. Our findings support the notion that cellular aging, and its reversal, might result from stochastic translation of metabolic inputs into resilient/plastic cell states via ER systems.

Functional consequences of pre- and postsynaptic expression of synaptic plasticity

Growing experimental evidence shows that both homeostatic and Hebbian synaptic plasticity can be expressed presynaptically as well as postsynaptically. In this review, we start by discussing this evidence and methods used to determine expression loci. Next, we discuss the functional consequences of this diversity in pre- and postsynaptic expression of both homeostatic and Hebbian synaptic plasticity. In particular, we explore the functional consequences of a biologically tuned model of pre- and postsynaptically expressed spike-timing-dependent plasticity complemented with postsynaptic homeostatic control. The pre- and postsynaptic expression in this model predicts (i) more reliable receptive fields and sensory perception, (ii) rapid recovery of forgotten information (memory savings), and (iii) reduced response latencies, compared with a model with postsynaptic expression only. Finally, we discuss open questions that will require a considerable research effort to better elucidate how the specific locus of expression of homeostatic and Hebbian plasticity alters synaptic and network computations.This article is part of the themed issue 'Integrating Hebbian and homeostatic plasticity'.

Formulation, construction and analysis of kinetic models of metabolism: A review of modelling frameworks

Kinetic models are critical to predict the dynamic behaviour of metabolic networks. Mechanistic kinetic models for large networks remain uncommon due to the difficulty of fitting their parameters. Recent modelling frameworks promise new ways to overcome this obstacle while retaining predictive capabilities. In this review, we present an overview of the relevant mathematical frameworks for kinetic formulation, construction and analysis. Starting with kinetic formalisms, we next review statistical methods for parameter inference, as well as recent computational frameworks applied to the construction and analysis of kinetic models. Finally, we discuss opportunities and limitations hindering the development of larger kinetic reconstructions.

Identifiability, reducibility, and adaptability in allosteric macromolecules

Bohner and Venkataraman propose a link between the sensitivity of allosteric macromolecules to their underlying biophysical parameters, the interrelationships between these parameters, and macromolecular adaptability. They argue that “emergent” combinations of parameters yield mechanistic insight that individual parameters cannot.

The ability of macromolecules to transduce stimulus information at one site into conformational changes at a distant site, termed “allostery,” is vital for cellular signaling. Here, we propose a link between the sensitivity of allosteric macromolecules to their underlying biophysical parameters, the interrelationships between these parameters, and macromolecular adaptability. We demonstrate that the parameters of a canonical model of the mSlo large-conductance Ca²⁺-activated K⁺ (BK) ion channel are non-identifiable with respect to the equilibrium open probability-voltage relationship, a common functional assay. We construct a reduced model with emergent parameters that are identifiable and expressed as combinations of the original mechanistic parameters. These emergent parameters indicate which coordinated changes in mechanistic parameters can leave assay output unchanged. We predict that these coordinated changes are used by allosteric macromolecules to adapt, and we demonstrate how this prediction can be tested experimentally. We show that these predicted parameter compensations are used in the first reported allosteric phenomena: the Bohr effect, by which hemoglobin adapts to varying pH.

Socio-ecological dynamics and challenges to the governance of Neglected Tropical Disease control

The current global attempts to control the so-called “Neglected Tropical Diseases (NTDs)” have the potential to significantly reduce the morbidity suffered by some of the world’s poorest communities. However, the governance of these control programmes is driven by a managerial rationality that assumes predictability of proposed interventions, and which thus primarily seeks to improve the cost-effectiveness of implementation by measuring performance in terms of pre-determined outputs. Here, we argue that this approach has reinforced the narrow normal-science model for controlling parasitic diseases, and in doing so fails to address the complex dynamics, uncertainty and socio-ecological context-specificity that invariably underlie parasite transmission. We suggest that a new governance approach is required that draws on a combination of non-equilibrium thinking about the operation of complex, adaptive, systems from the natural sciences and constructivist social science perspectives that view the accumulation of scientific knowledge as contingent on historical interests and norms, if more effective control approaches sufficiently sensitive to local disease contexts are to be devised, applied and managed. At the core of this approach is an emphasis on the need for a process that assists with the inclusion of diverse perspectives, social learning and deliberation, and a reflexive approach to addressing system complexity and incertitude, while balancing this flexibility with stability-focused structures. We derive and discuss a possible governance framework and outline an organizational structure that could be used to effectively deal with the complexity of accomplishing global NTD control. We also point to examples of complexity-based management structures that have been used in parasite control previously, which could serve as practical templates for developing similar governance structures to better manage global NTD control. Our results hold important wider implications for global health policy aiming to effectively control and eradicate parasitic diseases across the world.

Mathematical Justification of Expression-Based Pathway Activation Scoring (PAS)

Although modeling of activation kinetics for various cell signaling pathways has reached a high grade of sophistication and thoroughness, most such kinetic models still remain of rather limited practical value for biomedicine. Nevertheless, recent advancements have been made in application of signaling pathway science for real needs of prescription of the most effective drugs for individual patients. The methods for such prescription evaluate the degree of pathological changes in the signaling machinery based on two types of data: first, on the results of high-throughput gene expression profiling, and second, on the molecular pathway graphs that reflect interactions between the pathway members. For example, our algorithm OncoFinder evaluates the activation of molecular pathways on the basis of gene/protein expression data in the objects of the interest.Yet, the question of assessment of the relative importance for each gene product in a molecular pathway remains unclear unless one call for the methods of parameter sensitivity /stiffness analysis in the interactomic kinetic models of signaling pathway activation in terms of total concentrations of each gene product.Here we show two principal points: 1. First, the importance coefficients for each gene in pathways that were obtained using the extremely time- and labor-consuming stiffness analysis of full-scaled kinetic models generally differ from much easier-to-calculate expression-based pathway activation score (PAS) not more than by 30%, so the concept of PAS is kinetically justified. 2. Second, the use of pathway-based approach instead of distinct gene analysis, due to the law of large numbers, allows restoring the correlation between the similar samples that were examined using different transcriptome investigation techniques.

The Limitations of Model-Based Experimental Design and Parameter Estimation in Sloppy Systems

We explore the relationship among experimental design, parameter estimation, and systematic error in sloppy models. We show that the approximate nature of mathematical models poses challenges for experimental design in sloppy models. In many models of complex biological processes it is unknown what are the relevant physical mechanisms that must be included to explain system behaviors. As a consequence, models are often overly complex, with many practically unidentifiable parameters. Furthermore, which mechanisms are relevant/irrelevant vary among experiments. By selecting complementary experiments, experimental design may inadvertently make details that were ommitted from the model become relevant. When this occurs, the model will have a large systematic error and fail to give a good fit to the data. We use a simple hyper-model of model error to quantify a model’s discrepancy and apply it to two models of complex biological processes (EGFR signaling and DNA repair) with optimally selected experiments. We find that although parameters may be accurately estimated, the discrepancy in the model renders it less predictive than it was in the sloppy regime where systematic error is small. We introduce the concept of a sloppy system–a sequence of models of increasing complexity that become sloppy in the limit of microscopic accuracy. We explore the limits of accurate parameter estimation in sloppy systems and argue that identifying underlying mechanisms controlling system behavior is better approached by considering a hierarchy of models of varying detail rather than focusing on parameter estimation in a single model.

Sloppy models are often unidentifiable, i.e., characterized by many parameters that are poorly constrained by experimental data. Many models of complex biological systems are sloppy, which has prompted considerable debate about the identifiability of parameters and methods of selecting optimal experiments to infer parameter values. We explore how the approximate nature of models affects the prospect for accurate parameter estimates and model predictivity in sloppy models when using optimal experimental design. We find that sloppy models may no longer give a good fit to data generated from “optimal” experiments. In this case, the model has much less predictive power than it did before optimal experimental selection. We use a simple hyper-model of model error to quantify the model’s discrepancy from the physical system and discuss the potential limits of accurate parameter estimation in sloppy systems.

Conditions that Stabilize Membrane Domains Also Antagonize n-Alcohol Anesthesia

Diverse molecules induce general anesthesia with potency strongly correlated with both their hydrophobicity and their effects on certain ion channels. We recently observed that several n-alcohol anesthetics inhibit heterogeneity in plasma-membrane-derived vesicles by lowering the critical temperature (Tc) for phase separation. Here, we exploit conditions that stabilize membrane heterogeneity to further test the correlation between the anesthetic potency of n-alcohols and effects on Tc. First, we show that hexadecanol acts oppositely to n-alcohol anesthetics on membrane mixing and antagonizes ethanol-induced anesthesia in a tadpole behavioral assay. Second, we show that two previously described "intoxication reversers" raise Tc and counter ethanol's effects in vesicles, mimicking the findings of previous electrophysiological and behavioral measurements. Third, we find that elevated hydrostatic pressure, long known to reverse anesthesia, also raises Tc in vesicles with a magnitude that counters the effect of butanol at relevant concentrations and pressures. Taken together, these results demonstrate that ΔTc predicts anesthetic potency for n-alcohols better than hydrophobicity in a range of contexts, supporting a mechanistic role for membrane heterogeneity in general anesthesia.