Schedule for: 22w5155 - Mathematical Models in Biology: from Information Theory to Thermodynamics

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Many bacteria have the ability to form dormant endospores (“spores”) - a truly fascinating state of life. Spores can withstand harsh conditions and can survive without nutrients for many, many years. Compared to vegetative cells, their ability to resist common stress factors is enhanced tremendously while their metabolic activity is reduced by orders of magnitude, if there is any. Yet spores do monitor their environment; and when triggered appropriately, they will return into an active state and lose their resistance properties within minutes in a process called germination. The biophysics of how spores sense and respond to their environment is not well understood and experimentally challenging to address. Here I will describe my group’s efforts to establish a setup to study the response of individual spores to environmental changes. I will show preliminary evidence suggesting that dormant spores have a membrane potential that is tunable by internal and external factors. Our data also suggests that specific nutrients induce a proton motif force by triggering the spore’s germinant receptors. I will argue that a combination of single-spore experiments together with biophysical modelling and mutant studies provides a promising way to elucidate information processing by bacterial endospores.

The ability of biochemical systems to read and propagate molecular information critically depends on the ability of enzymes to discriminate among chemically similar substrates. I will describe how our theoretical framework lead to the new insights into kinetic proofreading (KPR), a mechanism that reduces errors below the equilibrium thermodynamic limit at the expense of increased energy dissipation and slower kinetics. Our results indicated how free-energy landscapes of the enzymes (E. coli ribosome, tRNA synthetase, and T7 DNA polymerase) evolved to optimize speed-accuracy-dissipation trade-offs in their underlying processes.

Biological sensory receptors provide perfect examples of microscopic scale information transduction in the presence of non-negligible thermal fluctuations. For example, studies of ligand-receptor reveal that accurate concentration sensing is achieved by integrating out noise in the sensor's stochastic trajectories. However, we argue that the stochastic trajectory is not always an adversary -- it could allow a single sensor to perform multiplexing (or muxing) by simultaneously transducing multiple environmental variables (e.g., concentration, temperature, and flow speed) to the downstream sensory network. This work develops a general theory of stochastic sensory muxing and a theoretical upper bound of muxing. The theory is demonstrated and verified by an exactly solvable Markov dynamics model, where an arbitrary sensor can achieve the upper bound of muxing without optimizing parameters. The theory is further demonstrated by a realistic Langevin dynamics simulation of a ligand receptor within a bath of ligands. Simulations verify that even a binary state ligand receptor with short-term memory can simultaneously sense two out of three independent environmental variables -- ligand concentration, temperature, and media's flow speed. Both models demonstrate that the upper bound for muxing is tight. This theory provides insights on designing novel microscopic sensors that are capable of muxing in realistic and complex environments.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The efficient coding hypothesis has revolutionized theoretical neuroscience. I would argue that its next-generation instantiation is best understood using rate-distortion theory, where rate can be justified as an energetic cost using lower bounds from stochastic thermodynamics. I use rate-distortion theory to inspire a simple model of sensory adaptation. In randomly drawn, fluctuating environments, this model explains the absence of sensory neurogenesis and predicts that biological sensors are poised to just barely confuse ``minimal confounds'' in the environment. Additional work, using other information-theoretic lower bounds on energetic costs, shows that the information-theoretic lower bounds used are not tight.

Information acquisition is crucial for cell survival, and many cellular sensors have evolved to be as precise as physically possible. Understanding these precision limits can therefore give important insights into the mechanisms and capabilities of cell sensing. First pioneered half a century ago in the context of bacterial chemotaxis, this way of viewing sensory biology has expanded to include concentration sensing, gradient sensing, flow sensing, mechanosensing, thermosensing, and more. I will discuss this history and describe some of our own work deriving and testing new physical bounds to sensory precision, including thermosensing by bacteria and self-guided flow sensing by cancer cells. This field reveals the fascinating physics that constrains cell behavior and suggests that cells operate at the edge of these physical bounds.

In addition to measuring chemical concentrations in their surroundings, cells must also infer concentrations within the cytoplasm and cellular compartments in order to maintain homeostasis. Measuring intracellular concentrations of chemical species fundamentally differs from measuring the extracellular environment due to the high degree of macromolecular crowding inside of cells and subcellular compartments. Crowding influences both the rate at which ligands and receptors encounter one another, as well as the transition rates between transient contacting ligand-receptor pairs and the stably bound complexes. Modeling such interactions poses two challenges: 1) The model must capture the heterogeneity of the crowded compartments, and 2) The simulation must be efficient enough to generate robust statistics of relatively rare binding events. In this work, we combine particle-based reaction diffusion simulations with Monte Carlo sampling to determine the accuracy of a concentration-sensing receptor as a function of macromolecular crowding at a significantly reduced computational cost. This method can be used to understand and quantify information processing by cellular signaling pathways in the crowded cellular interior.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Information is at the core of biology. Over a century of intense investigation, it has become evident that, in the most fundamental sense, biological processes have evolved to use organic matter to generate, preserve and transmit information. In a narrower sense, information is used to bias the distributions of states that can be adopted by biological systems. Thus, our understanding of these processes is shaped by the analysis of the states that the system can be in and the time scales over which the transitions between these states may occur. In this talk, I will discuss how the analysis of ergodicity in biological systems can reveal their controllability by informative events, suggesting a key framework for developing physical description of biological population dynamics.

The channel capacities of several mammalian signaling networks have been found to be much lower than what is observed. To address this discrepancy, we develop a new theoretical framework that explicitly accounts for intrinsic stochastic noise in signaling networks and extrinsic cell-to-cell variability when quantifying channel capacity. Using this framework, we estimate the channel capacity of two important mammalian pathways, the epidermal growth factor pathway, and the insulin-like growth factor pathway. Ultimately, our method leads to conceptually clearer and significantly higher estimates of channel capacities. We discuss the consequences for downstream cellular decision making.

(Co-presentation by Peter Thomas and Massimiliano Pierobon) Information transmission and storage have gained traction as unifying concepts to characterize biological systems and their chances of survival and evolution at multiple scales. Despite the potential for an information-based mathematical framework to offer new insights into life processes and ways to interact with and control them, the main legacy is that of Shannon’s, where a purely syntactic characterization of information scores systems on the basis of their maximum information efficiency. The latter metrics seem not entirely suitable for biological systems, where transmission and storage of different pieces of information (carrying different semantics) can result in different chances of survival. Based on an abstract mathematical model able to capture the parameters and behaviors of a population of single-celled organisms whose survival is correlated to information retrieval from the environment, this paper explores the aforementioned disconnect between classical information theory and biology. In this paper, we present a model, specified as a computational state machine, which is then utilized in a simulation framework constructed specifically to reveal emergence of a “subjective information”, i.e., trade-off between a living system’s capability to maximize the acquisi- tion of information from the environment, and the maximization of its growth and survival over time. Simulations clearly show that a strategy that maximizes information efficiency results in a lower growth rate with respect to the strategy that gains less information but contains a higher meaning for survival.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Being able to accurately predict a molecule's function from sequence alone would revolutionize fields from drug design to catalytic chemistry to agriculture and animal husbandry, but current methods are limited either by their ability to fit models (for regression-based techniques such as the Ising or Potts models) or else by the tendency to overfit (for machine-learning-based approaches). Here I present a new principled approach that avoids both the under- and over-fitting problems, by using an information-theoretic approach to the analysis of multiple-sequence alignments that can take multi-variable correlations (beyond order two) into account without any fitting whatsoever. Using typical thermodynamic constructions such as the Helmholtz free energy, this approach allows us to link information to energy, and create classifiers whose accuracy improves when higher-order correlations between the sequence's monomers are taken into account. If the size of the data set used to create the multiple-sequence alignment is small (the training set), we can prevent overfitting by simply limiting the order of correlations that is taken into account in the construction of the classifier. I show applications of this new tool to the functional classification of guide RNA sequences, proteins, as well as computer programs. The method is general so that it could also be used to predict what task a behaving animal is engaged in from the neuronal recordings only.

The relationship between genotype and phenotype defines evolution in the long run and human health day to day. Because it is rooted in random mutations, drift and opaque selection forces, this relationship seems intractably complex. It has a vast number of components, these form multiscale hierarchies of networks, and each network has cryptic emergent properties, seemingly precluding the type of simplifying, universal quantitative laws that are routinely found across physics and chemistry. An attractive solution might be to use statistical mechanics, hoping to model the broad macro properties of living systems free of the need to know any specific interaction details. In fact, analogies between statistical thermodynamics and population-genetic theories have been pointed out repeatedly, but these formalisms have lacked computable energy terms for organismal fitness and have remained abstract, with little ability to make predictions so far. We present here a new statistical mechanics of coding variants. This model leads to simple mathematical equations and statistical physics distributions that are fully computable and make testable predictions on the impact of protein mutations in fitness landscapes and on the genetic errors that foster complex human diseases, including cancer or Alzheimer’s. A question remains, however, whether the energetic and statistical physics interpretation of mutations and their ensembles is just an analogy — pleasant and fortuitous but ultimately misguided — or a fundamental principle of evolution whereby organisms maximize their mutational entropy whilst living within a narrow band of total mutational energy. To distinguish between these two possibilities, we turned to the theorem of equipartition. For equipartition to hold, we find that a term for the mutational inertia of each gene, or gene weight μ, that we did not know how to measure and had previously simply set uniformly to 1, now becomes fully specified and precisely computable. Strikingly, μ improves the fit of coding variants to statistical mechanics, and also identifies genes that are highly interconnected in protein-protein interaction networks; that mediate crucial homeostatic functions in the cell; that cause embryonic lethality or developmental abnormalities in the mouse; that are essential for bacterial survival or growth and that correlate with the functional impact of knockouts in E. coli. Finally, emerging data further show that μ identifies genes undergoing selection in various diseases. These data show that the theorem of equipartition predicts biological phenomena not included in our initial model, in further support that living systems evolve according to basic principles of statistical mechanics. In practice, the statistical physics of fitness landscape unifies molecular evolution and population genetics and may prove relevant to population risk stratification and to precision medicine.

Information thermodynamics relates the rate of change of mutual information between two interacting subsystems to their thermodynamics when the joined system is described by a bipartite stochastic dynamics satisfying local detailed balance. Here, we expand the scope of information thermodynamics to deterministic bipartite chemical reaction networks, namely, composed of two coupled subnetworks sharing species, but not reactions. We do so by introducing a meaningful notion of mutual information defined for non-normalized concentration distributions. This allows us to formulate separate second laws for each subnetwork, which account for their energy and information exchanges, in complete analogy with stochastic systems. We then use our framework to investigate the working mechanisms of a model of chemically-driven self-assembly and an experimental light-driven bimolecular motor. We show that both systems are constituted by two coupled subnetworks of chemical reactions. One subnetwork is maintained out of equilibrium by external reservoirs (chemostats or light sources) and powers the other via energy and information flows. In doing so, we clarify that the information flow is precisely the thermodynamic counterpart of an information ratchet mechanism only when no energy flow is involved.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Life depends on the input of energy, either directly provided by sunlight or in form of high- energy matter. The rules and conditions for the conversion of chemical or electromagnetic energy into living structure and all the processes related with life are governed by the laws of thermodynamics. Hence, to understand the potential and the limitations of cell growth and metabolism, it is unavoidable to take these laws into account. During the last years, systems biology has developed many mathematical models aiming to describe steady states and dynamic behavior of cellular processes in qualitative and quantitative terms. The validity of the model predictions depends strongly on whether the model formulation is in agreement with the laws of physics, chemistry, and, specifically, thermodynamics. Here, we review basic principles of thermodynamics for equilibrium and non-equilibrium processes as well as for closed and open systems as far as they concern processes of life and development. We illustrate the application of thermodynamic laws for some practical cases that are currently intensively studied in systems and computational biology. Specifically, we will discuss the concept of entropy production and energy dissipation for isolated and open systems and its interpretation for the feasibility of biological processes, especially metabolism. These findings are very important for biotechnological processes where energy dissipation should be limited.

Quorum sensing is a process through which bacteria can regulate gene expression according to their population density. The reasons for why bacteria use quorum sensing to regulate production of “public goods”, biochemical products that benefit nearby bacteria, are not entirely clear. We use mathematical modeling to explore how quorum sensing compares to other strategies for controlling production of public goods, namely unconditional production independent on population density, in small populations of bacteria where the random nature of growth is significant. Our model captures both how likely “cheater” strains, which do not produce public goods but benefit from them, are to take over a population and how long on average the population will last before going extinct. We find that depending on how expensive public good production is and how critical public goods are for growth, quorum sensing can decrease or increase the mean time to extinction compared with unconditional production, while always reducing the likelihood of cheaters taking over. Our results could have important implications for the growth of bacterial infections, for example Pseudomonas aeruginosa infections of the lungs of cystic fibrosis patients.

The chemical reaction networks that regulate living systems are all stochastic to varying degrees. The resulting randomness affects biological outcomes at multiple scales, from the probability that a single protein molecule successfully finds its folded state to the evolutionary trajectory of a population of cells. Understanding how the distribution of these outcomes changes over time is often difficult, and achieving control over this distribution via external interventions is an even more complex challenge. Intriguingly, this problem has close parallels in a very different domain: manipulating quantum states for applications like quantum computing and cold atom transport. In this talk we show how one can translate quantum control into the classical realm of biology, giving us a novel tool for steering biological processes. We illustrate this idea through two examples: the first is controlling the distribution of genetic variants in an evolving cellular population. This is motivated by recent efforts to combat antibiotic resistance via therapies that guide the evolution of pathogens toward maximized drug sensitivity. The second example involves controlling the distribution of protein folding states using so-called molecular chaperones: protein enzymes that facilitate the unfolding or disaggregating of misfolded proteins. The theoretical framework behind these two examples is quite general, and can in principle be used in many other biophysical problems. It also allows one to explore the thermodynamic costs associated with control. References: Iram, Dolson, Chiel et al., Nature Physics 17, 135 (2021); Ilker et al., Phys. Rev. X 12, 021048 (2022).

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Biological systems, ranging from the brain to the immune system, store memory of molecular interactions to efficiently recognize and respond to stimuli. However, the strategies to encode memory can vary largely across different systems. In this talk, I will discuss how statistics and dynamics of stimuli should determine the optimal memory encoding strategies in biological networks. In particular, I will contrast the compartmentalized memory in the adaptive immune system, which primarily interacts with evolving pathogens, with the distributed memory in the olfactory cortex, which interacts with relatively static odor molecules. Focusing on the adaptive immune system, I will discuss how memory encoding could be understood in light of host-pathogen coevolution. Specifically, I will show that to achieve a long-term benefit for the host, immune memory should be actively regulated, with a preference for cross-reactive receptors with a moderate affinity against pathogens as opposed to high affinity receptors. Our theory also predicts that an organism’s life-expectancy should strongly impact the cross-reactivity of its immune memory, and we expect organisms with shorter life expectancy to carry more cross-reactive memory. This theoretical prediction can guide more comprehensive cross-species comparisons of immune systems, which is currently missing from immunological studies.

The ability to predict the future is beneficial to many biological systems. For example, some organisms can react to an environment without glucose by making a costly adaptation to allow processing different types of sugars such as xylose. However, this is only worth doing if the cell can predict that glucose is unlikely to return. Alternatively, in bacterial chemotaxis, the cell must detect the gradient of resources it has experienced to predict whether an area of high resource density lies ahead. To make good predictions, a system must collect information about the current and past state of the environment and this information gathering also costs resources. In this talk we calculate the fundamental limits on the cost of extracting various useful pieces of information about a given environment and consider simple systems able to reach these limits. We discuss how a system can exploit correlations within the environment to reduce the costs of information gathering, and the consequences of the system having made an incorrect guess about the statistics of the environment.

Decision models based on normative principles have played a major role in our understanding of how the brain forms decisions. However, these models have typically been derived for simple, stable environments, and their relevance to decisions under more naturalistic, dynamic conditions is unclear. In this work, we derive normative models that balance the trade-off between information gathering and time cost to maximize performance (i.e., reward rate, or accuracy per unit time). Using dynamic programming, we find that normative models for stochastic environments exhibit non-monotonic dynamics that are well-tuned to environmental fluctuations. We also show that these adaptive normative models perform robustly even when implemented imperfectly (noisily). We conclude by modifying our analysis to incorporate a finite-energy budget in correlated decision environments. In this ongoing work, we demonstrate that normative models adaptively switch between gathering information and leveraging information from past decisions.

Evolutionary biology seeks to understand how genetic information changes over time to produce organisms that are well-adapted to their environments. While population genetics has developed a large body of theory regarding how genetic variance changes in the process of evolution, we lack correspondingly rich theory describing how adaptive information is acquired by the process of natural selection. The relative lack of rigorous treatments of genetic information is due in part to misconceptions about the meaning of this information and its connection to evolution by natural selection. First, we offer a formal definition of adaptive genetic information that integrates notions of correlational and causal information with a transmission sense of information that captures the inheritance and evolution of genetic information. Adaptive genetic information quantified in Shannon-theoretic terms refers to contents of the genome that represent selective features of particular environments. We show that this information is accumulated by divergent selection but not by non-adaptive processes such as drift, mutation, or gene flow in expectation. In addition, we demonstrate that this view of genetic information is not only of theoretical interest, but also offers a practical measure of adaptive differentiation. Next, we consider the relative effectiveness of natural selection as an information acquisition process. We reframe the ‘learning problem’ faced by an evolving population as a population versus environment (PvE) game, which can be applied to settings such as stochastic environments, frequency-dependent selection, and arbitrary environmental change. We show that the learning theoretic concept of ‘regret’ rigorously captures the efficiency of selection as a learning process. This lets us establish general bounds on the cost of information acquisition by natural selection, and we empirically validate these bounds in an experimental evolution system. We note that natural selection is a highly effective learning process in that selection is an asymptotically optimal algorithm for the problem faced by evolving populations, and no other algorithm can consistently outperform selection in general. Finally, we consider how this framework can be used to assess the value of adaptive therapies for treating cancer and other evolving burdens. Our results highlight the centrality of information to natural selection and the value of learning theory as a perspective on evolutionary biology.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk will present results for calculating the probabilities of rare events in models in Systems Biology within the framework of stochastic reaction networks. The theory of Large Deviations allows one to assess events whose probabilities are exponentially small and are hence not captured by Law of Large Numbers or the Central Limit Theorem results, yet are eventually likely to occur. We are particularly interested in dynamical observables of the models, which are time additive functionals of the stochastic processes describing the model dynamics, and are reasonably easy to access in experimental studies via empirical distributions. Long term behaviour of these functionals reveals their stability and equilibria, and the portion of time the dynamics of the process spends in different parts of the state space. Using tools for Markov processes we can describe the large deviation rate function for these dynamical observables, which involve numerically solving partial integro-differential equations with boundary values. We further present our results and their numerical implementation on an example of a system of chemical reactions with mean bistable dynamics. Our results are used to effectively compare the long run fit of two approximating processes modelling this system of reactions: a pure jump Markov process, and a reflected jump-diffusion process.

All-atom and coarse-grained molecular dynamics (MD), Langevin dynamics (LD) and Brownian dynamics (BD) are computational methodologies, which have been applied to spatio-temporal modelling of a number of intracellular processes. I will discuss connections between MD, LD and BD, with a focus on the development, analysis and applications of multi-resolution methods, which use (detailed) MD simulations in localized regions of particular interest (in which accuracy and microscopic details are important) and a (less-detailed) coarser stochastic model in other regions in which accuracy may be traded for simulation efficiency. I will discuss applications of multi-resolution methodologies to modelling of intracellular calcium dynamics, actin dynamics and DNA dynamics.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

We explore the effects of stochastic resetting to random positions of a Brownian particle on first passage times and Shannon's entropy. We explore the different entropy regimes, namely, the externally-driven, the zero-entropy and the Maxwell demon regimes. We show that the mean first passage time (MFPT) minimum can be found in any of these regimes. We provide a novel analytical method to compute the MFPT, the mean first passage number of resets (MFPNR) and mean first passage entropy (MFPE) in the case where the Brownian particle resets to random positions sampled from a set of distributions known a priori. We show the interplay between the reset position distribution's second moment and the reset rate, and the effect it has on the MFPT and MFPE. We further propose a mechanism whereby the entropy per reset can be either in the Maxwell demon or the externally driven regime, yet the overall mean first passage entropy corresponds to the zero-entropy regime. Additionally, we find an overlap between the dynamic phase space and the entropy phase space. We use this method in a generalized version of the Evans-Majumdar model by assuming the reset position is random and sampled from a Gaussian distribution. We then consider the toggling reset whereby the Brownian particle resets to a random position sampled from a distribution dependent on the reset parity. All our results are compared to and in agreement with numerical simulations.

Cells are the basic units of life. Although every cell in the body have essentially the same genetic code, chromatin states in the genome regulate different genes and allow distinct cells to perform specific functions. Recently, new sequencing techniques have begun to map these chromatin states at the single-cell level, opening up new opportunities to infer gene regulatory principles at unprecedented resolutions. I will talk about integrating these technologies with statistical frameworks to reveal global insights in chromatin regulation in single cells. I will present new technologies that map histone modifications in single cells, and show how to model the high-dimensional sparse count data to understand gene regulation. Specifically, I infer transcription factor activities in single cells, revealing how blood stem cells rewire regulatory networks to become distinct mature blood cell types. I will also show how integrating statistical and experimental frameworks can infer multiple histone modifications in single cells. I apply this framework to reveal hierarchical structure of chromatin regulation during blood formation and chromatin velocity. Overall, combining models into sequencing technologies reveals global gene regulatory principles.