Principles of adaptive sorting revealed by in silico evolution

Reliable ligand discrimination in stochastic multistep kinetic proofreading: First passage time vs. product counting strategies

Cellular signaling, crucial for biological processes like immune response and homeostasis, relies on specificity and fidelity in signal transduction to accurately respond to stimuli amidst biological noise. Kinetic proofreading (KPR) is a key mechanism enhancing signaling specificity through time-delayed steps, although its effectiveness is debated due to intrinsic noise potentially reducing signal fidelity. In this study, we reformulate the theory of kinetic proofreading (KPR) by convolving multiple intermediate states into a single state and then define an overall "processing" time required to traverse these states. This simplification allows us to succinctly describe kinetic proofreading in terms of a single waiting time parameter, facilitating a more direct evaluation and comparison of KPR performance across different biological contexts such as DNA replication and T cell receptor (TCR) signaling. We find that loss of fidelity for longer proofreading steps relies on the specific strategy of information extraction and show that in the first-passage time (FPT) discrimination strategy, longer proofreading steps can exponentially improve the accuracy of KPR at the cost of speed. Thus, KPR can still be an effective discrimination mechanism in the high noise regime. However, in a product concentration-based discrimination strategy, longer proofreading steps do not necessarily lead to an increase in performance. However, by introducing activation thresholds on product concentrations, can we decompose the product-based strategy into a series of FPT-based strategies to better resolve the subtleties of KPR-mediated product discrimination. Our findings underscore the importance of understanding KPR in the context of how information is extracted and processed in the cell.

Proofreading does not result in more reliable ligand discrimination in receptor signaling due to its inherent stochasticity

Kinetic proofreading (KPR) has been used as a paradigmatic explanation for the high specificity of ligand discrimination by cellular receptors. KPR enhances the difference in the mean receptor occupancy between different ligands compared to a nonproofread receptor, thus potentially enabling better discrimination. On the other hand, proofreading also attenuates the signal and introduces additional stochastic receptor transitions relative to a nonproofreading receptor. This increases the relative magnitude of noise in the downstream signal, which can interfere with reliable ligand discrimination. To understand the effect of noise on ligand discrimination beyond the comparison of the mean signals, we formulate the task of ligand discrimination as a problem of statistical estimation of the receptor affinity of ligands based on the molecular signaling output. Our analysis reveals that proofreading typically worsens ligand resolution compared to a nonproofread receptor. Furthermore, the resolution decreases further with more proofreading steps under most commonly biologically considered conditions. This contrasts with the usual notion that KPR universally improves ligand discrimination with additional proofreading steps. Our results are consistent across a variety of different proofreading schemes and metrics of performance, suggesting that they are inherent to the KPR mechanism itself rather than any particular model of molecular noise. Based on our results, we suggest alternative roles for KPR schemes such as multiplexing and combinatorial encoding in multi-ligand/multi-output pathways.

How the T cell signaling network processes information to discriminate between self and agonist ligands

T cells exhibit remarkable sensitivity and selectivity in detecting and responding to agonist peptides (p) bound to MHC molecules in a sea of self pMHC molecules. Despite much work, understanding of the underlying mechanisms of distinguishing such ligands remains incomplete. Here, we quantify T cell discriminatory capacity using channel capacity, a direct measure of the signaling network's ability to discriminate between antigen-presenting cells (APCs) displaying either self ligands or a mixture of self and agonist ligands. This metric shows how differences in information content between these two types of peptidomes are decoded by the topology and rates of kinetic proofreading signaling steps inside T cells. Using channel capacity, we constructed numerically substantiated hypotheses to explain the discriminatory role of a recently identified slow LAT Y132 phosphorylation step. Our results revealed that in addition to the number and kinetics of sequential signaling steps, a key determinant of discriminatory capability is spatial localization of a minimum number of these steps to the engaged TCR. Biochemical and imaging experiments support these findings. Our results also reveal the discriminatory role of early negative feedback and necessary amplification conferred by late positive feedback.

Reliable target ligand detection by noise-induced receptor cluster formation

Intracellular reactions are intrinsically stochastic. Nonetheless, cells can reliably respond to the changing environment by sensing their target molecules sensitively and specifically, even with the existence of abundant structurally-similar non-target molecules. The mechanism of how the cells can balance and achieve such different characteristics is not yet fully understood. In this work, we demonstrate that these characteristics can be attained by a ligand-induced stochastic cluster formation of receptors via the noise-induced symmetry breaking, in which the intrinsic stochasticity works to enhance sensitivity and specificity. We also show that the noise-induced cluster formation enables cells to detect the target ligand reliably by compensating the abundant non-target ligands in the environment. The proposed mechanism may lead to a deeper understanding of a biological function of the receptor clustering and provide an alternative candidate for the reliable ligand detection to the kinetic proofreading.

Light-based tuning of ligand half-life supports kinetic proofreading model of T cell signaling

T cells are thought to discriminate self from foreign peptides by converting small differences in ligand binding half-life into large changes in cell signaling. Such a kinetic proofreading model has been difficult to test directly, as existing methods of altering ligand binding half-life also change other potentially important biophysical parameters, most notably the mechanical stability of the receptor-ligand interaction. Here we develop an optogenetic approach to specifically tune the binding half-life of a chimeric antigen receptor without changing other binding parameters and provide direct evidence of kinetic proofreading in T cell signaling. This half-life discrimination is executed in the proximal signaling pathway, downstream of ZAP70 recruitment and upstream of diacylglycerol accumulation. Our methods represent a general tool for temporal and spatial control of T cell signaling and extend the reach of optogenetics to probe pathways where the individual molecular kinetics, rather than the ensemble average, gates downstream signaling.

In Silico Evolution of Biochemical Log-Response

Numerous biological systems are known to harbor a form of logarithmic behavior, from Weber's law to bacterial chemotaxis. Such a log-response allows for sensitivity to small relative variations of biochemical inputs over a large range of concentration values. Here we use a genetic algorithm to evolve biochemical networks displaying a logarithmic response. A quasi-perfect log-response implemented by the same core network evolves in a convergent way across our different in silico replications. The best network is able to fit a logarithm over 4 orders of magnitude with an accuracy of the order of 1%. At the heart of this network, we show that a logarithmic approximation may be implemented with one single nonlinear interaction, that can be interpreted either as multisite phosphorylations or as a ligand induced multimerization. We provide an analytical explanation for the effect and exhibit constraints on parameters. Biological log-response might thus be easier to implement than usually assumed.

Receptor crosstalk improves concentration sensing of multiple ligands

Cells need to reliably sense external ligand concentrations to achieve various biological functions such as chemotaxis or signaling. The molecular recognition of ligands by surface receptors is degenerate in many systems, leading to crosstalk between ligand-receptor pairs. Crosstalk is often thought of as a deviation from optimal specific recognition, as the binding of noncognate ligands can interfere with the detection of the receptor's cognate ligand, possibly leading to a false triggering of a downstream signaling pathway. Here we quantify the optimal precision of sensing the concentrations of multiple ligands by a collection of promiscuous receptors. We demonstrate that crosstalk can improve precision in concentration sensing and discrimination tasks. To achieve superior precision, the additional information about ligand concentrations contained in short binding events of the noncognate ligand should be exploited. We present a proofreading scheme to realize an approximate estimation of multiple ligand concentrations that reaches a precision close to the derived optimal bounds. Our results help rationalize the observed ubiquity of receptor crosstalk in molecular sensing.

Identifying feasible operating regimes for early T-cell recognition: The speed, energy, accuracy trade-off in kinetic proofreading and adaptive sorting

In the immune system, T cells can quickly discriminate between foreign and self ligands with high accuracy. There is evidence that T-cells achieve this remarkable performance utilizing a network architecture based on a generalization of kinetic proofreading (KPR). KPR-based mechanisms actively consume energy to increase the specificity beyond what is possible in equilibrium. An important theoretical question that arises is to understand the trade-offs and fundamental limits on accuracy, speed, and dissipation (energy consumption) in KPR and its generalization. Here, we revisit this question through numerical simulations where we simultaneously measure the speed, accuracy, and energy consumption of the KPR and adaptive sorting networks for different parameter choices. Our simulations highlight the existence of a "feasible operating regime" in the speed-energy-accuracy plane where T-cells can quickly differentiate between foreign and self ligands at reasonable energy expenditure. We give general arguments for why we expect this feasible operating regime to be a generic property of all KPR-based biochemical networks and discuss implications for our understanding of the T cell receptor circuit.

φ-evo: A program to evolve phenotypic models of biological networks

Molecular networks are at the core of most cellular decisions, but are often difficult to comprehend. Reverse engineering of network architecture from their functions has proved fruitful to classify and predict the structure and function of molecular networks, suggesting new experimental tests and biological predictions. We present φ-evo, an open-source program to evolve in silico phenotypic networks performing a given biological function. We include implementations for evolution of biochemical adaptation, adaptive sorting for immune recognition, metazoan development (somitogenesis, hox patterning), as well as Pareto evolution. We detail the program architecture based on C, Python 3, and a Jupyter interface for project configuration and network analysis. We illustrate the predictive power of φ-evo by first recovering the asymmetrical structure of the lac operon regulation from an objective function with symmetrical constraints. Second, we use the problem of hox-like embryonic patterning to show how a single effective fitness can emerge from multi-objective (Pareto) evolution. φ-evo provides an efficient approach and user-friendly interface for the phenotypic prediction of networks and the numerical study of evolution itself.

Untangling the Hairball: Fitness-Based Asymptotic Reduction of Biological Networks

Complex mathematical models of interaction networks are routinely used for prediction in systems biology. However, it is difficult to reconcile network complexities with a formal understanding of their behavior. Here, we propose a simple procedure (called ϕ¯) to reduce biological models to functional submodules, using statistical mechanics of complex systems combined with a fitness-based approach inspired by in silico evolution. The ϕ¯ algorithm works by putting parameters or combination of parameters to some asymptotic limit, while keeping (or slightly improving) the model performance, and requires parameter symmetry breaking for more complex models. We illustrate ϕ¯ on biochemical adaptation and on different models of immune recognition by T cells. An intractable model of immune recognition with close to a hundred individual transition rates is reduced to a simple two-parameter model. The ϕ¯ algorithm extracts three different mechanisms for early immune recognition, and automatically discovers similar functional modules in different models of the same process, allowing for model classification and comparison. Our procedure can be applied to biological networks based on rate equations using a fitness function that quantifies phenotypic performance.

Balancing specificity, sensitivity, and speed of ligand discrimination by zero-order ultraspecificity

Specific interactions between receptors and their target ligands in the presence of nontarget ligands are crucial for biological processes such as T cell ligand discrimination. To discriminate between the target and nontarget ligands, cells have to increase specificity to the target ligands by amplifying the small differences in affinity among ligands. In addition, sensitivity to the ligand concentration and quick discrimination are also important to detect low amounts of target ligands and facilitate fast cellular decision making after ligand recognition. In this work we propose a mechanism for nonlinear specificity amplification (ultraspecificity) based on zero-order saturating reactions, which was originally proposed to explain nonlinear sensitivity amplification (ultrasensitivity) to the ligand concentration. In contrast to the previously proposed proofreading mechanisms that amplify the specificity by a multistep reaction, our model can produce an optimal balance of specificity, sensitivity, and quick discrimination. Furthermore, we show that a model for insensitivity to a large number of nontarget ligands can be naturally derived from a model with the zero-order ultraspecificity. The zero-order ultraspecificity, therefore, may provide an alternative way to understand ligand discrimination from the viewpoint of nonlinear properties in biochemical reactions.

Combining in silico evolution and nonlinear dimensionality reduction to redesign responses of signaling networks

The rational design of network behavior is a central goal of synthetic biology. Here, we combine in silico evolution with nonlinear dimensionality reduction to redesign the responses of fixed-topology signaling networks and to characterize sets of kinetic parameters that underlie various input-output relations. We first consider the earliest part of the T cell receptor (TCR) signaling network and demonstrate that it can produce a variety of input-output relations (quantified as the level of TCR phosphorylation as a function of the characteristic TCR binding time). We utilize an evolutionary algorithm (EA) to identify sets of kinetic parameters that give rise to: (i) sigmoidal responses with the activation threshold varied over 6 orders of magnitude, (ii) a graded response, and (iii) an inverted response in which short TCR binding times lead to activation. We also consider a network with both positive and negative feedback and use the EA to evolve oscillatory responses with different periods in response to a change in input. For each targeted input-output relation, we conduct many independent runs of the EA and use nonlinear dimensionality reduction to embed the resulting data for each network in two dimensions. We then partition the results into groups and characterize constraints placed on the parameters by the different targeted response curves. Our approach provides a way (i) to guide the design of kinetic parameters of fixed-topology networks to generate novel input-output relations and (ii) to constrain ranges of biological parameters using experimental data. In the cases considered, the network topologies exhibit significant flexibility in generating alternative responses, with distinct patterns of kinetic rates emerging for different targeted responses.

Phenotypic spandrel: absolute discrimination and ligand antagonism

We consider the general problem of sensitive and specific discrimination between biochemical species. An important instance is immune discrimination between self and not-self, where it is also observed experimentally that ligands just below the discrimination threshold negatively impact response, a phenomenon called antagonism. We characterize mathematically the generic properties of such discrimination, first relating it to biochemical adaptation. Then, based on basic biochemical rules, we establish that, surprisingly, antagonism is a generic consequence of any strictly specific discrimination made independently from ligand concentration. Thus antagonism constitutes a 'phenotypic spandrel': a phenotype existing as a necessary by-product of another phenotype. We exhibit a simple analytic model of discrimination displaying antagonism, where antagonism strength is linear in distance from the detection threshold. This contrasts with traditional proofreading based models where antagonism vanishes far from threshold and thus displays an inverted hierarchy of antagonism compared to simpler models. The phenotypic spandrel studied here is expected to structure many decision pathways such as immune detection mediated by TCRs and FCϵRIs, as well as endocrine signalling/disruption.

Intrinsic limits to gene regulation by global crosstalk

Gene regulation relies on the specificity of transcription factor (TF)–DNA interactions. Limited specificity may lead to crosstalk: a regulatory state in which a gene is either incorrectly activated due to noncognate TF–DNA interactions or remains erroneously inactive. As each TF can have numerous interactions with noncognate cis-regulatory elements, crosstalk is inherently a global problem, yet has previously not been studied as such. We construct a theoretical framework to analyse the effects of global crosstalk on gene regulation. We find that crosstalk presents a significant challenge for organisms with low-specificity TFs, such as metazoans. Crosstalk is not easily mitigated by known regulatory schemes acting at equilibrium, including variants of cooperativity and combinatorial regulation. Our results suggest that crosstalk imposes a previously unexplored global constraint on the functioning and evolution of regulatory networks, which is qualitatively distinct from the known constraints that act at the level of individual gene regulatory elements.

Limited specificity of transcription factor-DNA interactions leads to crosstalk in gene regulation. Here the authors consider global crosstalk in regulatory networks of growing size and complexity, and show that it imposes constraints on gene regulation and on the evolution of regulatory networks.

Positional information from oscillatory phase shifts : insights from in silico evolution

Complex cellular decisions are based on temporal dynamics of pathways, including genetic oscillators. In development, recent works on vertebrae formation have suggested that relative phase of genetic oscillators encode positional information, including differentiation front defining vertebrae positions. Precise mechanisms for this are still unknown. Here, we use computational evolution to find gene network topologies that can compute the phase difference between oscillators and convert it into a decoder morphogen concentration. Two types of networks are discovered, based on symmetry properties of the decoder gene. So called asymmetric networks are studied, and two submodules are identified converting phase information into an amplitude variable. Those networks naturally display a 'shock' for a well defined phase difference, that can be used to define a wavefront of differentiation. We show how implementation of these ideas reproduce experimental features of vertebrate segmentation.

Enzyme Sequestration as a Tuning Point in Controlling Response Dynamics of Signalling Networks

Signalling networks result from combinatorial interactions among many enzymes and scaffolding proteins. These complex systems generate response dynamics that are often essential for correct decision-making in cells. Uncovering biochemical design principles that underpin such response dynamics is a prerequisite to understand evolved signalling networks and to design synthetic ones. Here, we use in silico evolution to explore the possible biochemical design space for signalling networks displaying ultrasensitive and adaptive response dynamics. By running evolutionary simulations mimicking different biochemical scenarios, we find that enzyme sequestration emerges as a key mechanism for enabling such dynamics. Inspired by these findings, and to test the role of sequestration, we design a generic, minimalist model of a signalling cycle, featuring two enzymes and a single scaffolding protein. We show that this simple system is capable of displaying both ultrasensitive and adaptive response dynamics. Furthermore, we find that tuning the concentration or kinetics of the sequestering protein can shift system dynamics between these two response types. These empirical results suggest that enzyme sequestration through scaffolding proteins is exploited by evolution to generate diverse response dynamics in signalling networks and could provide an engineering point in synthetic biology applications.

Physical Limit to Concentration Sensing Amid Spurious Ligands

To adapt their behavior in changing environments, cells sense concentrations by binding external ligands to their receptors. However, incorrect ligands may bind nonspecifically to receptors, and when their concentration is large, this binding activity may interfere with the sensing of the ligand of interest. Here, I derive analytically the physical limit to the accuracy of concentration sensing amid a large number of interfering ligands. A scaling transition is found when the mean bound time of correct ligands is twice that of incorrect ligands. I discuss how the physical bound can be approached by a cascade of receptor states generalizing kinetic proofreading schemes.

BioJazz: in silico evolution of cellular networks with unbounded complexity using rule-based modeling

Systems biologists aim to decipher the structure and dynamics of signaling and regulatory networks underpinning cellular responses; synthetic biologists can use this insight to alter existing networks or engineer de novo ones. Both tasks will benefit from an understanding of which structural and dynamic features of networks can emerge from evolutionary processes, through which intermediary steps these arise, and whether they embody general design principles. As natural evolution at the level of network dynamics is difficult to study, in silico evolution of network models can provide important insights. However, current tools used for in silico evolution of network dynamics are limited to ad hoc computer simulations and models. Here we introduce BioJazz, an extendable, user-friendly tool for simulating the evolution of dynamic biochemical networks. Unlike previous tools for in silico evolution, BioJazz allows for the evolution of cellular networks with unbounded complexity by combining rule-based modeling with an encoding of networks that is akin to a genome. We show that BioJazz can be used to implement biologically realistic selective pressures and allows exploration of the space of network architectures and dynamics that implement prescribed physiological functions. BioJazz is provided as an open-source tool to facilitate its further development and use. Source code and user manuals are available at: http://oss-lab.github.io/biojazz and http://osslab.lifesci.warwick.ac.uk/BioJazz.aspx.

Chemodetection in fluctuating environments: receptor coupling, buffering, and antagonism

Variability in the chemical composition of the extracellular environment can significantly degrade the ability of cells to detect rare cognate ligands. Using concepts from statistical detection theory, we formalize the generic problem of detection of small concentrations of ligands in a fluctuating background of biochemically similar ligands binding to the same receptors. We discover that in contrast with expectations arising from considerations of signal amplification, inhibitory interactions between receptors can improve detection performance in the presence of substantial environmental variability, providing an adaptive interpretation to the phenomenon of ligand antagonism. Our results suggest that the structure of signaling pathways responsible for chemodetection in fluctuating and heterogeneous environments might be optimized with respect to the statistics and dynamics of environmental composition. The developed formalism stresses the importance of characterizing nonspecific interactions to understand function in signaling pathways.

Insights into the initiation of TCR signaling

The initiation of T cell antigen receptor signaling is a key step that can result in T cell activation and the orchestration of an adaptive immune response. Early events in T cell receptor signaling can distinguish between agonist and endogenous ligands with exquisite selectivity, and show extraordinary sensitivity to minute numbers of agonists in a sea of endogenous ligands. We review our current knowledge of models and crucial molecules that aim to provide a mechanistic explanation for these observations. Building on current understanding and a discussion of unresolved issues, we propose a molecular model for initiation of T cell receptor signaling that may serve as a useful guide for future studies.

Graphlet signature-based scoring method to estimate protein-ligand binding affinity

Over the years, various computational methodologies have been developed to understand and quantify receptor-ligand interactions. Protein-ligand interactions can also be explained in the form of a network and its properties. The ligand binding at the protein-active site is stabilized by formation of new interactions like hydrogen bond, hydrophobic and ionic. These non-covalent interactions when considered as links cause non-isomorphic sub-graphs in the residue interaction network. This study aims to investigate the relationship between these induced sub-graphs and ligand activity. Graphlet signature-based analysis of networks has been applied in various biological problems; the focus of this work is to analyse protein-ligand interactions in terms of neighbourhood connectivity and to develop a method in which the information from residue interaction networks, i.e. graphlet signatures, can be applied to quantify ligand affinity. A scoring method was developed, which depicts the variability in signatures adopted by different amino acids during inhibitor binding, and was termed as GSUS (graphlet signature uniqueness score). The score is specific for every individual inhibitor. Two well-known drug targets, COX-2 and CA-II and their inhibitors, were considered to assess the method. Residue interaction networks of COX-2 and CA-II with their respective inhibitors were used. Only hydrogen bond network was considered to calculate GSUS and quantify protein-ligand interaction in terms of graphlet signatures. The correlation of the GSUS with pIC50 was consistent in both proteins and better in comparison to the Autodock results. The GSUS scoring method was better in activity prediction of molecules with similar structure and diverse activity and vice versa. This study can be a major platform in developing approaches that can be used alone or together with existing methods to predict ligand affinity from protein-ligand complexes.

Noise and information transmission in promoters with multiple internal States

Based on the measurements of noise in gene expression performed during the past decade, it has become customary to think of gene regulation in terms of a two-state model, where the promoter of a gene can stochastically switch between an ON and an OFF state. As experiments are becoming increasingly precise and the deviations from the two-state model start to be observable, we ask about the experimental signatures of complex multistate promoters, as well as the functional consequences of this additional complexity. In detail, we i), extend the calculations for noise in gene expression to promoters described by state transition diagrams with multiple states, ii), systematically compute the experimentally accessible noise characteristics for these complex promoters, and iii), use information theory to evaluate the channel capacities of complex promoter architectures and compare them with the baseline provided by the two-state model. We find that adding internal states to the promoter generically decreases channel capacity, except in certain cases, three of which (cooperativity, dual-role regulation, promoter cycling) we analyze in detail.

Evolving phenotypic networks in silico

Evolved gene networks are constrained by natural selection. Their structures and functions are consequently far from being random, as exemplified by the multiple instances of parallel/convergent evolution. One can thus ask if features of actual gene networks can be recovered from evolutionary first principles. I review a method for in silico evolution of small models of gene networks aiming at performing predefined biological functions. I summarize the current implementation of the algorithm, insisting on the construction of a proper "fitness" function. I illustrate the approach on three examples: biochemical adaptation, ligand discrimination and vertebrate segmentation (somitogenesis). While the structure of the evolved networks is variable, dynamics of our evolved networks are usually constrained and present many similar features to actual gene networks, including properties that were not explicitly selected for. In silico evolution can thus be used to predict biological behaviours without a detailed knowledge of the mapping between genotype and phenotype.