Information Processing, Noise, and Adaptation in Living Systems

Unicellular organisms possess the remarkable capacity to grow and proliferate in vastly different environments by adapting their metabolism. While the benefits of this metabolic adaptation are self-evident, its costs remain largely unknown. Because growth occurs far from thermodynamic equilibrium, free-energy dissipation has long been hypothesized to pose key constraints to cellular life and its evolution. To investigate this hypothesis, we curated over 500 unicellular growth experiments across evolution and environments and analyzed them using a minimal non-equilibrium thermodynamic framework. We find that despite vast variability in biomass yield and dissipation, the dissipative cost of growth is largely conserved across metabolic types. This conserved cost constraint arises from a fine balance between the large amounts of free energy flowing in and out of cells and is independent of microbial species. Our work places non-equilibrium thermodynamics at the center of cellular growth and proliferation and paves the way for future investigations on the interplay between metabolism, evolution, and non-equilibrium thermodynamics.

[Replacement for the corresponding talk by Prof. Marc Timme (TU Dresden)]

Propagating, storing and processing information is key to take smart decisions – for organisms as well as for autonomous devices. In search for the minimal units that allow for complex behaviour the slime mould Physarum polycephalum stands out by solving complex optimization problems despite its simple make-up. Physarum’s body is an interlaced network of fluid-filled tubes lacking any nervous system, in fact being a single gigantic cell. Yet, Physarum finds the shortest path through a maze. We unravel that Physarum’s complex behaviour emerges from the physics of active flows shuffling through its tubular networks. Flows transport information, information that is stored in the architecture of the network. Thus, tubular adaptation drives processing of information into complex behaviour.

Living systems from biochemical networks to neural networks are responsible for carrying out precise biological functions and information processing tasks. However, most of these complex networks operate far out of equilibrium in which equilibrium statistical mechanics fails to describe even their steady state properties. It thus remains a major challenge in biological physics to develop a theoretical framework for studying these highly nonequilibrium complex systems. A central problem in living (or active) systems is how they manage to perform vital functions (e.g., adaptation, replication, and computing, etc.) accurately by using highly noisy signals. What are the mechanisms to control noise for accurate information processing? What are the energy costs for implementing these mechanisms? What are the design principles for achieving these biological functions efficiently? In the past 16 years, we have been working to answer these questions in various biochemical systems including ultra-sensitive switch, sensory adaptation, accurate biochemical oscillations, and collective behaviors (e.g., synchronization, flocking, and pattern formation) by using tools and concepts from nonequilibrium statistical physics. In this talk, we will first describe the general background on the topic followed by presenting some of our work related to the role of energy dissipation for important biological functions for sensory adaptation and (if time permits) synchronization of molecular clocks.

Biological processes are complex and involve many components whose interactions are stochastic and only partially characterized. Instead of ignoring or guessing unknown details our work focuses on analyzing classes of biochemical reaction networks that share some features but are left to vary arbitrarily in all unknown features. This allows us to derive impossibility constraints that can guide the design of synthetic biology applications and help us understand the operating principles of cellular processes. Additionally, I will discuss how invariants for classes of incompletely specified systems can be exploited to infer rate functions and causal interactions in complex systems from observing stochastic fluctuations of a small subset of components. Such approaches have the potential to turn quantitatively intractable complex systems into a sequence of solvable inference problems.

I will discuss our recent work (10.1103/PhysRevLett.131.077101) on the asymmetry of cross-correlations, i.e., steady-state correlations between different observables at two different points in time. As first shown by Onsager, a defining feature of equilibrium is that cross-correlations are symmetric under exchange of observables. We show, on the other hand, that the violation of this symmetry provides a bound on the thermodynamic cycle affinity, reflecting the strength of nonequilibrium driving. Our result leads to thermodynamic tradeoffs for various biomolecular functions that depend on such asymmetry. For example, our result implies a thermodynamic bound on the coherence of noisy oscillations, which was previously conjectured by Barato and Seifert. As another example, our result implies a thermodynamic bound on directed information flow in biochemical signal transduction. Finally, the approach provides an experimentally accessible way to infer thermodynamic affinity from empirical measurements.

I present an autonomous model of a chemical Maxwell demon that works by rectifying thermal fluctuations of chemical reactions. It constitutes the analog of a recently studied electronic demon. I describe its performances and its scaling behavior in the macroscopic limit. Moreover, I discuss the information thermodynamics of the Maxwell demon from a bipartite perspective.

Starting from structural and biochemical data, we show that ABC transporters are a billion years old realisation of autonomius Maxwell demons, whose function can be interpreted in terms of measurement, feedback and reset precisely as stated by Landauer and Bennet.

Biological systems must survive in highly variable environments with strong recurrent and random fluctuations. The ability to anticipate and adapt to such variability is critical to perform the intended biological functions. I will present a general theory for incorporating forecasting into the biological context of continuous adaptation. It demonstrates that predictivity generally allows adaptation to the optimal state with fewer resources and slower rates than in its absence. The results show that widely used mathematical approaches, such as linear or non-linear extrapolation, cannot be reliably implemented in the noisy context of biological systems. In contrast, mechanisms that identify recurrent fluctuations, such as the periodic programs of circadian rhythms, outperform naïve adaptation in most typical conditions.

Living systems are maintained out-of-equilibrium by external driving forces, thus continuously dissipating energy. Non-equilibrium steady states are characterized by emergent selection phenomena that break equilibrium symmetries dictated solely by energetic properties. In our recent work, we use the matrix-tree theorem to derive universal thermodynamic bounds on these symmetry-breaking features in biochemical systems. The presented bounds are independent of the kinetics and hold for both closed and open reaction networks, whether they are uni-molecular or catalytic. Our results can also be extended to the Master Equations in the chemical space. Using our framework, we recover the thermodynamic constraints in kinetic proofreading. Finally, we show that the contrast of reaction-diffusion patterns can be bounded only by the non-equilibrium driving force. The generality of these results paves the way to understanding the potentialities and limitations of non-equilibrium conditions in shaping steady-state properties of biochemical systems.

Systems like the prototypical lac operon can reliably hold the repression of transcription upon DNA replication across cell cycles with just ten repressor molecules per cell and, in addition, behave as if they were at equilibrium. The origin of this type of phenomena is still an unresolved question of major implications. Here, we develop a general theory to analyze strong perturbations in quasi-equilibrium systems and use it to quantify the effects of DNA replication in gene regulation. We find a scaling law that connects actual transcription with its predicted equilibrium values in terms of a single kinetic parameter. We show that even the simplest, exceptionally reliable natural system functions beyond the physical limits of naive regulation through compensatory mechanisms that suppress nonequilibrium effects. We validate the approach with both in vivo cell-population and single-cell characterization of the lac operon. Analyses of synthetic systems without adjuvant activators, such as the cAMP receptor protein (CRP), do not show this reliability. Our results provide a rationale for the function of CRP, beyond just being a tunable activator, as a mitigator of cell cycle perturbations.

Biological system process information despite noise-corrupted input, often operating at physical limits. A prime example is chemotaxis, i.e., active navigation of biological cells in spatial fields of chemical cues. Intriguingly, cells of different size use two different chemotaxis strategies, comparing concentrations in either space or time. Only heuristic arguments exist to explain this evolutionary choice. We present an information theory of an ideal agent that combines both strategies to quantify ‘chemotaxis in bits’ [1]. This enables us to predict when each strategy provides more information as function of a new powerlaw that combines agent size, motility noise and sensing noise. We demonstrate our theory with a bio-inspired search robot. [1] https://www.biorxiv.org/content/10.1101/2023.10.14.562229v1

Organisms, ranging from bacteria to animals, can predict changes in their environment. To this end, they need to encode the past signal into the dynamics of the internal system from which they predict the future. Yet, storing information is costly while not all features of the past signal are equally predictive. Here, we show for cellular systems that the bits of past information that are most informative about the future are also prohibitively costly, in terms of physical resources like protein copies and energy. Consequently, systems that maximize predictive power under a resource constraint differ from those that maximize this under an information-compression constraint. These findings have implications for the optimal design of both natural and man-made information-processing systems.

Biological and neural networks are adaptive - their connections slowly change in response to the state of the coupled elements making up the systems. The dynamics of such adaptive networks are intriguingly complex, rendering it extremely difficult to answer the fundamental question of what the requirements for maintaining functionally robust collective states under environmental stochasticity are. Aiming to understand the essential role of noise in robust adaptation, a basic problem in evolution and learning, we present a general statistical-physics theory for such dynamical systems, thus contributing to the development of an exact analytical treatment of the nonequilibrium phase transitions in large-scale multiple-time systems with network adaptation. As an illustration of our theory, we apply it to biological evolution, where phenotypes are shaped by gene-expression fast dynamics that are subjected to an external noise while genotypes are encoded by the configurations of a network of gene regulations. This network slowly evolves under natural selection with a mutation rate, depending on how adapted the shaped phenotypes are. Here we show how, for high reciprocity of network interactions, a robust phase, where both genotype and phenotype achieve optimal values, can emerge from a non-robust one via a non-equilibrium phase transition within an intermediate level of external noise. We also establish an analytical relation between the averaged entropy production rate (EPR) and the evolutionary speed which is quantified by the genetic variance. Specifically, EPR is shown to increase with increasing evolutionary speed at a given noise level.

Development relies on the ability of cells to organize into patterns of different cell types that underlie the formation of tissues and organs, with or without the use of external cues. Such patterning occurs in a reproducible manner despite the inevitable presence of noise. However, how to generically quantify the patterning performance of different developmental systems has remained unclear. We start by reviewing our past work that formalized positional information during externally-cued patterning in the language of bits. Our new work extends this concept to a wide range of self-organized developmental models. A novel information-theoretic measure for self-organized fate specification during embryonic development assesses the total information content of cell fate patterns, and decomposes it into an interpretable positional (old) and correlational (new) contribution. By optimizing the proposed measure, our framework provides a normative theory for developmental circuits, which we demonstrate on lateral inhibition, cell type proportioning, and reaction-diffusion models of self-organization. This paves a way towards a classification of developmental systems based on a common information-theoretic language, thereby organizing the zoo of implicated chemical and mechanical signaling processes.

Complex systems are characterized by multiple spatial and temporal scales. A natural framework to capture their multiscale nature is that of multilayer networks, where different layers represent distinct physical processes that often regulate each other indirectly. We model these regulatory mechanisms through triadic higher-order interactions between nodes and edges. This is especially relevant in the case of biochemical signaling networks, which constitute the fundamental building blocks for information processing in biological and living systems. In this work, we focus on how the different timescales associated with each layer impact their effective couplings in terms of their mutual information. We unravel the general principles governing how such information propagates across the multiscale structure, and apply them to study archetypal examples of biological signaling networks and effective environmental dependencies in stochastic processes. Our framework generalizes to any dynamics on multilayer networks, paving the way for a deeper understanding of how the multiscale nature of real-world systems shapes their information content and complexity.

How much can we learn about a stimulus by studying the activity of a sensory system encoding it? An important step to answer this question is to identify the amount of (mutual) information (MI) that is conveyed by a neural population in a certain time, the mutual information rate (MI rate). Because MI is constituted by dynamic processes during a finite time in our case (for example the movement of an object and a series of spikes) we define the MI rate as MI/∆t with ∆t non-infinitesimal. In general, it will depend on ∆t and also on the size of the time bin dt the neural signal is discretized with. These dependencies are informative about the time scales of the stimulus and the neurons processing it. The challenge in treating them emerges from the correlation of the activity in the k=∆t/dt different time bins inside the observation time. Actually, the complexity of exactly computing the mutual information grows exponentially in k, even for a single neuron. We solve this problem using approximations from statistical field theory adapted to the computation of entropies of many interacting units (Kühn & van Wijland 2023). For inhomogeneous Poisson spiking, the MI rate quickly drops with ∆t for k = 1; the larger we choose k, the slower this drop becomes. However, the dependence of the MI rate on ∆t is not always monotonous. Introducing a refractory period, e.g., decorrelates the response. This effect is transiently enhanced by increasing ∆t (leaving k fixed), leading to an increase of the MI rate. Further enlarging ∆t lets the MI rate drop again because too large chunks of the signal get lumped into one bin. We therefore detect an optimal temporal scale in terms of the MI rate - depending on the respective scales of neurons and stimuli, which down-stream areas might adapt to. More generally, because our method requires a number of estimates growing only quadratically in k, we are confident that it will prove useful for the study of the interaction of time scales shaping data from electrophysiological recordings.

How neuronal subtypes are generated at distinct times and proportions during human retinal development is poorly understood. Here, we investigated how blue cone photoreceptors develop before red/green cones in human retinal organoids. We find that DIO3, an enzyme that degrades thyroid hormone (TH), is a master regulator of cone developmental timing and stability. DIO3 is highly expressed in progenitor cells and decreases as cells differentiate into neurons, lowering TH degradation over time. Loss of DIO3 activity leads to precocious blue and red/green cone development, increased cone density, and conversion of blue cones to red/green fate. Feedback in the TH signaling pathway ensures homeostatic expression of regulators. In a computational model, we find that the increase in TH levels upon neuronal differentiation coupled with TH-dependent specification provide robustness to cone development compared to an intrinsic model without autoregulation. Our work suggests that differentiation and signaling are coordinated to specify neuronal subtypes in the human retina.

Cells express genes in order to respond to environmental changes, differentiate or decide their fates, and develop into a healthy organism. Gene expression is regulated by cues, such as changing transcription factor concentrations, whose concentrations are often low. This expression in response to a changing concentration can be viewed as a type of decision that can be analyzed in terms of an information-theoretic framework. In this talk, I will show, on the example of early fly embryo development, how such an information-theoretic inference approach can help us understand features of a complex transcriptional apparatus that may be difficult to model, due to the complexity of the contributing regulatory factors. I will compare the inferred optimal sensor to past, heuristic models for regions on the DNA that respond to transcription factors, and, relate their architecture to features commonly found in efficient computing systems.

Signal processing in biochemical networks relies on dynamic informa- tion transmission between time-trajectories of the respective molecular components. From a mathematical point of view, the transferred in- formation can be described via the mutual information. Traditionally, this has been calculated using a Gaussian approximation. Our re- cent work suggests that this method is not always suitable for every biochemical networks which can lead to quantitative and qualitative mismatches to the exact solution. In this work, we explain the origin of these discrepancies and present a modified version of the Gaussian framework that aligns better with the characteristics of stochastic bio- chemical networks.

Diabetes, a disease characterized by high blood glucose, is one of the most common chronic diseases in the developed world. Systems biology modeling of diabetes — and endocrine disorders more generally — is a field where quantitative biologists have not been active until recently. I will present two sets of questions in this field, which have excited us. First, I will argue that ketosis-prone diabetes, a subtype of the disease that has features of both type 1 and type 2 diabetes, requires us to rethink the basic physiology of pancreatic beta cells, suggesting existence of new beta-cell phenotypes, which are waiting to be discovered. Second, I will argue that the pancreatic endocrine circuit is too complex for the textbook description of its function, and that viewing it as a multi-objective controller may open new ways to understanding its structure and failure modes. While both of these snippets are very preliminary, I will try to convince you that they already make it clear that physics-style thinking is essential to address these and related questions, and to open a new page in quantitative understanding of human endocrine system.

Signaling pathways are commonly seen as linear or branched biochemical conduits of information. Their analysis involves signals traveling between specific input and output points. However, signaling networks in eukaryotic cells are usually more complex, with substantial crosstalk between pathways. The pathway cross-activation can lead to interference between information channels, and signal modulation. In this talk, I will show experimental evidence suggesting that this type of crosstalk can define the ‘noise’ in the NF-kappaB signaling pathway, which emerges from the mechanical micro-environment of the cell. In particular, I will discuss the biochemical interplay between the NF-kappaB and YAP pathways in different cell types, under different environmental conditions. This analysis reveals the roots and consequences of pathway cross-modulation allowing coupling between key signaling inputs and outputs, including those reporting on mechanical and biochemical microenvironment states.

In various biological systems information from many noisy molecular receptors must be integrated into a collective response. A striking example is the thermal imaging organ of pit vipers. Single nerve fibers in the organ reliably respond to mK temperature increases, a thousand times more sensitive than their molecular sensors, thermo-TRP ion channels. We propose a mechanism for the integration of this molecular information. In our model, amplification arises due to proximity to a dynamical bifurcation, separating a regime with frequent and regular action potentials (APs), from a regime where APs are irregular and infrequent. Near the transition, AP frequency can have an extremely sharp dependence on temperature, naturally accounting for the thousand-fold amplification. Furthermore, close to the bifurcation, most of the information about temperature available in the TRP channels’ kinetics can be read out from the times between consecutive APs even in the presence of readout noise. A key model prediction is that the coefficient of variation in the distribution of interspike times decreases with AP frequency, and quantitative comparison with experiments indeed suggests that nerve fibers of snakes are located very close to the bifurcation. While proximity to such bifurcation points typically requires fine-tuning of parameters, we propose that having feedback act from the order parameter (AP frequency) onto the control parameter robustly maintains the system in the vicinity of the bifurcation. This robustness suggests that similar feedback mechanisms might be found in other sensory systems which also need to detect tiny signals in a varying environment.

Organisms use sensory inputs to perform survival-relevant behaviors. Sometimes these tasks are difficult, in the sense that the organism’s ability to accomplish them is noisy or unreliable. What makes these tasks difficult? Here, using a combination of theory and single-cell experiments, we study how the fidelity of sensory information affects the ability of E. coli bacteria to climb up chemical gradients. In the first part, we ask how the information a cell can obtain bounds its gradient-climbing speed. We derive a theoretical speed limit and then measure cells’ chemotaxis speeds and information rates. Our results show that cells efficiently use the little information they have, climbing gradients at speeds near the information-performance bound. In the second part, we ask what limits the information cells get during chemotaxis. We derive the information limit imposed by the physics of counting diffusing ligand molecules, and we experimentally measure how cells compare to this limit. We find that E. coli are far from the physical limits on sensing—instead, their information about external signals is limited by noise in the internal components they use to perform computations. Thus, information matters for bacterial chemotaxis, and this task is difficult for E. coli, not because of fundamental limits on sensing, but because of other constraints on their signaling systems. These results shed light on successes and constraints in the evolution of bacteria’s ability to bias their motion towards favorable regions.

Complex lifeforms host microbiota, microbes that live synergistically with their host. Accordingly, hosts have mechanisms to defend against and tolerate the microbiota. The intestinal mucus, where these systems collide, plays a pivotal role in managing this relationship, yet lacks an integrative theoretical framework. We propose a minimal model to elucidate dynamics at this interface, focusing on the ileum's mucus defense. The model considers the effect of delay in host antimicrobial peptide secretion and how the host can use two different signals, from the bulk microbiota and from segmented filamentous bacteria (SFB). Our theory suggests the host can optimize defense by minimizing antimicrobial peptide production and controlling bacterial exposure. Integrating two recent experiments, we show host dynamics are consistent with sensing both bulk and SFB, supporting our 'optimal defense' hypothesis. Therefore, we propose that similar mechanisms could prove advantageous to other species and applicable beyond the ileum's mucus barrier.

Arthur Genthon, Carl Modes, Stephan Grill, Frank Jülicher How to accurately copy information from a polymer template sequence? What is the energetic cost of this accuracy? To address these questions, we consider the replication of a polymer template via the active template-directed assembly and disassembly of monomeric building blocks, which consumes fuel energy (ATP). This process competes with the passive spontaneous assembly and disassembly of monomers, which produces random polymer sequences. We obtain a phase diagram for the accuracy of the polymer copies, measured by their mean monomer error fraction. A non-equilibrium phase transition between accurate and inaccurate copies of the template is observed when the driving fuel energy is varied. In this transition, the fuel energy threshold depends on the complexity (entropy) of the polymer and is thus akin to Landauer’s principle. A cost-accuracy trade-off is observed as a function of the number of monomer types, where the accuracy of information transmission is maximised for small number of monomer types. The impact of kinetic proofreading on these results is also discussed.

Behavioral strategies employed for chemotaxis have been studied across phyla, but the neural computations underlying navigational decisions remain elusive. By combining electrophysiology, quantitative behavioral analysis and computational modeling, we explore how olfactory signals experienced during free motion are processed by the olfactory system of the Drosophila larva. We create virtual sensory environments based on optogenetics to study how peripheral olfactory information is converted into elementary orientation responses. Our work aims to unravel the loop that organizes the acquisition of sensory information through active movements of the body.

Animals often use odors to send messages that travel across long distances, carried by the wind. At the receiver’s end, the signal arrives attenuated and mangled by the turbulent atmospheric flow. Nevertheless, insects are able to decode the sequence of sparse odor encounters and use this information about the source location to build a successful search strategy. A key role in this process is played by the memory that the insect has to keep about the history of previous odor detections. Here, we show in a computational model of olfactory search that finite-state controllers, very simple algorithmic devices endowed with a minimal memory, are rich enough to explain the occurrence of several behavioral patterns that are indeed observed in nature.

Harvesting free energy from the environment is essential for the operation of many biological and artificial systems. We investigate the maximum rate of harvesting achievable by optimizing a set of reactions in a Markovian system, possibly given topological, kinetic, and thermodynamic constraints. We show that the maximum harvesting rate can be expressed as a variational principle, which we solve in closed-form for three physically meaningful regimes. Our approach is relevant for optimal design and for quantifying efficiency of existing reactions. Our results are illustrated on bacteriorhodopsin, a light-driven proton pump from Archae, which is found to be close to optimal under realistic conditions.

Biomolecular condensates provide distinct chemical environments, which can control various cellular processes. The fluorescent labeling of molecules enables molecular tracking and provides an invaluable tool to probe key processes in cell biology. We discuss how biomolecular condensates govern the kinetics of chemical reactions and how this is reflected in the dynamics of labeled molecules. Our theoretical approach provides insights into how the dynamics of labeled molecules can be used to determine chemical reaction rates inside and outside bimolecular condensates. Finally, we address single-molecule trajectories and relate their statistics to the physics of phase separation.

Perceptual decision-making frequently requires making rapid, reliable choices upon encountering noisy sensory inputs accumulated as evidence. Neural recordings from the lateral intra-parietal area in humans and primates performing perceptual decision-making tasks highlighted evidence accumulation mechanisms, often modeled as drifted diffusions. In most of such experimental approaches, decision-makers were exposed to computer-generated randomly fluctuating stimuli whose motion is not linked to any physical phenomenon. To better characterize the statistical processes underlying human decision-making, we performed behavioral experiments with human participants visualizing fluctuations of physical nonequilibrium stationary states and analyzed their responses in the light of stochastic thermodynamics. Participants were asked to watch hundreds of movies of a particle endowed with drifted Brownian dynamics and to judge the motion as leftward or rightward quickly and reliably. Our results uncovered fundamental performance limits consistent with recently established thermodynamic trade-offs (uncertainty relations, TURs) involving speed, accuracy, and dissipation. Specifically, they reveal that mean decision times are inversely proportional to the entropy production rate. Moreover, to achieve a given level of observed accuracy, participants require more time than predicted by Wald’s optimal sequential probability ratio test, indicating suboptimal integration of available information. In view of such suboptimality, we develop an alternative account equipped with non-Markovian evidence integration with a memory time constant. Our results suggest that humans adapt their memory integration window to the rate of dissipation of the observed phenomenon, favoring memory over momentary evidence for effective decisions in scenarios where stimuli are far from equilibrium. Overall, our study shows that perceptual psychophysics using stimuli rooted in nonequilibrium physical processes provides a robust platform for understanding how the human brain makes decisions on stochastic information inputs.

The advent of novel optogenetics technology allows the recording of brain activity with a resolution never seen before. The characterization of these very large data sets offers new challenges as well as unique theory-testing opportunities. Here we discuss whether the spatial and temporal correlations of the collective activity of thousands of neurons are tangled as predicted by the theory of critical phenomena. The analysis shows that both the correlation length $\xi$ and the correlation time $\tau$ scale as predicted as a function of the system size. With some peculiarities that we discuss, the analysis uncovers evidence consistent with the view that the large-scale brain cortical dynamics corresponds to critical phenomena.

Synchronization in the brain is often present in an intermittent fashion, i.e., there are intervals of synchronized activity interspersed with intervals of desynchronized activity. A series of experimental studies showed that this kind of temporal patterning of neural synchronization may be very specific and may be correlated with behavior (even if the average synchrony strength is not changed). Prior studies showed that a network with many short desynchronized intervals may be functionally different from a network with few long desynchronized intervals as it may be more sensitive to synchronizing input signals. We use biologically motivated neural circuits models to understand how cellular properties, network organization, and noise may regulate the temporal patterning of phase-locked episodes in the network activity and help the networks to adapt to input signals.

Selection and variation are both key aspects in the evolutionary process. Previous research on the mapping between molecular sequence (genotype) and molecular fold (phenotype) has shown the presence of several structural properties in different biological contexts, implying that these might be universal in evolutionary spaces. The deterministic genotype–phenotype (GP) map that links short RNA sequences to minimum free energy secondary structures has been studied extensively because of its computational tractability and biologically realistic nature. However, this mapping ignores the phenotypic plasticity of RNA. We define a GP map that incorporates non-deterministic (ND) phenotypes, and take RNA as a case study; we use the Boltzmann probability distribution of folded structures and examine the structural properties of ND GP maps for RNA sequences of length 12 and coarse-grained RNA structures of length 30 (RNAshapes30). A framework is presented to study robustness, evolvability and neutral spaces in the ND map. This framework is validated by demonstrating close correspondence between the ND quantities and sample averages of their deterministic counterparts. When using the ND framework we observe the same structural properties as in the deterministic GP map, such as bias, negative correlation between genotypic robustness and evolvability, and positive correlation between phenotypic robustness and evolvability.

Kalman Filter (KF) is an optimal linear state prediction algorithm, with applications in fields as diverse as engineering, economics, robotics, and space exploration. Here, we develop an extension of the KF, called a Pathspace Kalman Filter (PKF) which allows us to a) dynamically track the uncertainties associated with the underlying data and prior knowledge, and b) take as input an entire trajectory and an underlying mechanistic model, and using a Bayesian methodology quantify the different sources of uncertainty. An application of this algorithm is to automatically detect temporal windows where the internal mechanistic model deviates from the data in a time-dependent manner. First, we provide theorems characterizing the convergence of the PKF algorithm. Then, we numerically demonstrate that the PKF outperforms conventional KF methods on a synthetic dataset lowering the mean-squared-error by several orders of magnitude. Finally, we apply this method to biological time-course dataset involving over 1.8 million gene expression measurements.