Cell-to-Cell Heterogeneity in Trypanosomes.

Recent developments in single-cell and single-molecule techniques have revealed surprising levels of heterogeneity among isogenic cells. These advances have transformed the study of cell-to-cell heterogeneity into a major area of biomedical research, revealing that it can confer essential advantages, such as priming populations of unicellular organisms for future environmental stresses. Protozoan parasites, such as trypanosomes, face multiple and often hostile environments, and to survive, they undergo multiple changes, including changes in morphology, gene expression, and metabolism. But why does only a subset of proliferative cells differentiate to the next life cycle stage? Why do only some bloodstream parasites undergo antigenic switching while others stably express one variant surface glycoprotein? And why do some parasites invade an organ while others remain in the bloodstream? Building on extensive research performed in bacteria, here we suggest that biological noise can contribute to the fitness of eukaryotic pathogens and discuss the importance of cell-to-cell heterogeneity in trypanosome infections.

Stochasticity may generate coherent motion in bird flocks.

Murmurations along with other forms of flocking have come to epitomize collective animal movements. Most studies into these stunning aerial displays have aimed to understand how coherent motion may emerge from simple behavioral rules and behavioral correlations. These studies may now need revision because recently it has been shown that flocking birds, like swarming insects, behave on the average as if they are trapped in elastic potential wells. Here I show, somewhat paradoxically, how coherent motion can be generated by variations in the intensity of multiplicative noise which causes the shape of a potential well to change, thereby shifting the positions and strengths of centres of attraction. Each bird, irrespective of its position in the flock will respond in a similar way to such changes, giving the impression that the flock behaves as one, and typically resulting in scale-free correlations. I thereby show how correlations can be an emergent property of noisy, confining potential wells. I also show how such wells can lead to high density borders, a characteristic of flocks, and I show how they can account for the complex patterns of collective escape patterns of starling flocks under predation. I suggest swarming and flocking do not constitute two distinctly different kinds of collective behavior but rather that insects are residing in relatively stable potential wells whilst birds are residing in unstable potential wells. It is shown how, dependent upon individual perceptual capabilities, bird flocks can be poised at criticality.

How to correctly quantify neuronal phase-response curves from noisy recordings.

At the level of individual neurons, various coding properties can be inferred from the input-output relationship of a cell. For small inputs, this relation is captured by the phase-response curve (PRC), which measures the effect of a small perturbation on the timing of the subsequent spike. Experimentally, however, an accurate experimental estimation of PRCs is challenging. Despite elaborate measurement efforts, experimental PRC estimates often cannot be related to those from modeling studies. In particular, experimental PRCs rarely resemble the characteristic theoretical PRC expected close to spike initiation, which is indicative of the underlying spike-onset bifurcation. Here, we show for conductance-based model neurons that the correspondence between theoretical and measured phase-response curve is lost when the stimuli used for the estimation are too large. In this case, the derived phase-response curve is distorted beyond recognition and takes on a generic shape that reflects the measurement protocol and masks the spike-onset bifurcation. We discuss how to identify appropriate stimulus strengths for perturbation and noise-stimulation methods, which permit to estimate PRCs that reliably reflect the spike-onset bifurcation - a task that is particularly difficult if a lower bound for the stimulus amplitude is dictated by prominent intrinsic neuronal noise.

On signalling and estimation limits for molecular birth-processes.

Understanding and uncovering the mechanisms or motifs that molecular networks employ to regulate noise is a key problem in cell biology. As it is often difficult to obtain direct and detailed insight into these mechanisms, many studies instead focus on assessing the best precision attainable on the signalling pathways that compose these networks. Molecules signal one another over such pathways to solve noise regulating estimation and control problems. Quantifying the maximum precision of these solutions delimits what is achievable and allows hypotheses about underlying motifs to be tested without requiring detailed biological knowledge. The pathway capacity, which defines the maximum rate of transmitting information along it, is a widely used proxy for precision. Here it is shown, for estimation problems involving elementary yet biologically relevant birth-process networks, that capacity can be surprisingly misleading. A time-optimal signalling motif, called birth-following, is derived and proven to better the precision expected from the capacity, provided the maximum signalling rate constraint is large and the mean one above a certain threshold. When the maximum constraint is relaxed, perfect estimation is predicted by the capacity. However, the true achievable precision is found highly variable and sensitive to the mean constraint. Since the same capacity can map to different combinations of rate constraints, it can only equivocally measure precision. Deciphering the rate constraints on a signalling pathway may therefore be more important than computing its capacity.

Limits of noise for autoregulated gene expression.

Gene expression is influenced by extrinsic noise (involving a fluctuating environment of cellular processes) and intrinsic noise (referring to fluctuations within a cell under constant environment). We study the standard model of gene expression including an (in-)active gene, mRNA and protein. Gene expression is regulated in the sense that the protein feeds back and either represses (negative feedback) or enhances (positive feedback) its production at the stage of transcription. While it is well-known that negative (positive) feedback reduces (increases) intrinsic noise, we give a precise result on the resulting fluctuations in protein numbers. The technique we use is an extension of the Langevin approximation and is an application of a central limit theorem under stochastic averaging for Markov jump processes (Kang et al. in Ann Appl Probab 24:721-759, 2014). We find that (under our scaling and in equilibrium), negative feedback leads to a reduction in the Fano factor of at most 2, while the noise under positive feedback is potentially unbounded. The fit with simulations is very good and improves on known approximations.

Noise decomposition of intracellular biochemical signaling networks using nonequivalent reporters.

Experimental measurements of biochemical noise have primarily focused on sources of noise at the gene expression level due to limitations of existing noise decomposition techniques. Here, we introduce a mathematical framework that extends classical extrinsic-intrinsic noise analysis and enables mapping of noise within upstream signaling networks free of such restrictions. The framework applies to systems for which the responses of interest are linearly correlated on average, although the framework can be easily generalized to the nonlinear case. Interestingly, despite the high degree of complexity and nonlinearity of most mammalian signaling networks, three distinct tumor necrosis factor (TNF) signaling network branches displayed linearly correlated responses, in both wild-type and perturbed versions of the network, across multiple orders of magnitude of ligand concentration. Using the noise mapping analysis, we find that the c-Jun N-terminal kinase (JNK) pathway generates higher noise than the NF-κB pathway, whereas the activation of c-Jun adds a greater amount of noise than the activation of ATF-2. In addition, we find that the A20 protein can suppress noise in the activation of ATF-2 by separately inhibiting the TNF receptor complex and JNK pathway through a negative feedback mechanism. These results, easily scalable to larger and more complex networks, pave the way toward assessing how noise propagates through cellular signaling pathways and create a foundation on which we can further investigate the relationship between signaling system architecture and biological noise.

Linear noise approximation for stochastic oscillations of intracellular calcium.

A stochastic model of intracellular calcium oscillations is analytically studied. The governing master equation is expanded under the linear noise approximation and a closed prediction for the power spectrum of fluctuations analytically derived. A peak in the obtained power spectrum profile signals the presence of stochastic, noise induced oscillations which extend also outside the region where a deterministic limit cycle is predicted to occur.

Stochastic noise reduction upon complexification: positively correlated birth-death type systems.

Cell systems consist of a huge number of various molecules that display specific patterns of interactions, which have a determining influence on the cell׳s functioning. In general, such complexity is seen to increase with the complexity of the organism, with a concomitant increase of the accuracy and specificity of the cellular processes. The question thus arises how the complexification of systems - modeled here by simple interacting birth-death type processes - can lead to a reduction of the noise - described by the variance of the number of molecules. To gain understanding of this issue, we investigated the difference between a single system containing molecules that are produced and degraded, and the same system - with the same average number of molecules - connected to a buffer. We modeled these systems using Ito stochastic differential equations in discrete time, as they allow straightforward analytical developments. In general, when the molecules in the system and the buffer are positively correlated, the variance on the number of molecules in the system is found to decrease compared to the equivalent system without a buffer. Only buffers that are too noisy themselves tend to increase the noise in the main system. We tested this result on two model cases, in which the system and the buffer contain proteins in their active and inactive state, or protein monomers and homodimers. We found that in the second test case, where the interconversion terms are non-linear in the number of molecules, the noise reduction is much more pronounced; it reaches up to 20% reduction of the Fano factor with the parameter values tested in numerical simulations on an unperturbed birth-death model. We extended our analysis to two arbitrary interconnected systems, and found that the sum of the noise levels in the two systems generally decreases upon interconnection if the molecules they contain are positively correlated.

Signatures of nonlinearity in single cell noise-induced oscillations.

A class of theoretical models seeks to explain rhythmic single cell data by postulating that they are generated by intrinsic noise in biochemical systems whose deterministic models exhibit only damped oscillations. The main features of such noise-induced oscillations are quantified by the power spectrum which measures the dependence of the oscillatory signal's power with frequency. In this paper we derive an approximate closed-form expression for the power spectrum of any monostable biochemical system close to a Hopf bifurcation, where noise-induced oscillations are most pronounced. Unlike the commonly used linear noise approximation which is valid in the macroscopic limit of large volumes, our theory is valid over a wide range of volumes and hence affords a more suitable description of single cell noise-induced oscillations. Our theory predicts that the spectra have three universal features: (i) a dominant peak at some frequency, (ii) a smaller peak at twice the frequency of the dominant peak and (iii) a peak at zero frequency. Of these, the linear noise approximation predicts only the first feature while the remaining two stem from the combination of intrinsic noise and nonlinearity in the law of mass action. The theoretical expressions are shown to accurately match the power spectra determined from stochastic simulations of mitotic and circadian oscillators. Furthermore it is shown how recently acquired single cell rhythmic fibroblast data displays all the features predicted by our theory and that the experimental spectrum is well described by our theory but not by the conventional linear noise approximation.

A Frequency-Dependent Dynamic Electric-Mechanical Network for Thin-Wafer Piezoelectric Transducers Polarized in the Thickness Direction: Physical Model and Experimental Confirmation.

This paper is concerned with electric-acoustic/acoustic-electric conversions of thin-wafer piezoelectric transducers polarized in the thickness direction. By introducing two mechanical components with frequency-dependent values, i.e., radiation resistance and radiation mass, into the equivalent circuit of the thin-wafer piezoelectric transducer, we established a frequency-dependent dynamic mechanic-electric equivalent network with four terminals for an arbitrary given frequency, an enhancement from the conventional circuit networks. We derived the analytic expressions of its electric-acoustic and acoustic-electric conversion impulse responses using the four-terminal equivalent circuit to replace the traditional six-terminal equivalent circuit for a thin-wafer transducer with harmonic vibrational motion. For multifrequency electrical/acoustic signals acting on the transducer, we established parallel electric-acoustic/acoustic-electric conversion transmission networks. These two transmission network models have simple structures and clear physical and mathematical descriptions of thin-wafer transducers for electric-acoustic/acoustic-electric conversion when excited by a multifrequency electric/acoustic signal wavelet. The calculated results showed that the transducer's center frequency shift relates to its mechanical load and vibration state. The method reported in this paper can be applied to conventional-sized and small-sized piezoelectric transducers with universal applicability.

Efficient Quantification of Extrinsic Fluctuations via Stochastic Simulations.

At the molecular level, all the biological processes are exposed to fluctuations emanating from various sources in and around the cellular system. Often these fluctuations dictate the outcome of a cell-fate decision-making event. Thus, having an accurate estimate of these fluctuations for any biological network is extremely important. There are well-established theoretical and numerical methods to quantify the intrinsic fluctuation present within a biological network arising due to the low copy numbers of cellular components. Unfortunately, the extrinsic fluctuations arising due to cell division events, epigenetic regulation, etc. have received very little attention. However, recent studies demonstrate that these extrinsic fluctuations significantly affect the transcriptional heterogeneity of certain important genes. Herein, we propose a new stochastic simulation algorithm to efficiently estimate these extrinsic fluctuations for experimentally constructed bidirectional transcriptional reporter systems along with the intrinsic variability. We use the Nanog transcriptional regulatory network and its variants to illustrate our numerical method. Our method reconciled experimental observations related to Nanog transcription, made exciting predictions, and can be applied to quantify intrinsic and extrinsic fluctuations for any similar transcriptional regulatory network.

Deep networks may capture biological behavior for shallow, but not deep, empirical characterizations.

We assess whether deep convolutional networks (DCN) can account for a most fundamental property of human vision: detection/discrimination of elementary image elements (bars) at different contrast levels. The human visual process can be characterized to varying degrees of "depth," ranging from percentage of correct detection to detailed tuning and operating characteristics of the underlying perceptual mechanism. We challenge deep networks with the same stimuli/tasks used with human observers and apply equivalent characterization of the stimulus-response coupling. In general, we find that popular DCN architectures do not account for signature properties of the human process. For shallow depth of characterization, some variants of network-architecture/training-protocol produce human-like trends; however, more articulate empirical descriptors expose glaring discrepancies. Networks can be coaxed into learning those richer descriptors by shadowing a human surrogate in the form of a tailored circuit perturbed by unstructured input, thus ruling out the possibility that human-model misalignment in standard protocols may be attributable to insufficient representational power. These results urge caution in assessing whether neural networks do or do not capture human behavior: ultimately, our ability to assess "success" in this area can only be as good as afforded by the depth of behavioral characterization against which the network is evaluated. We propose a novel set of metrics/protocols that impose stringent constraints on the evaluation of DCN behavior as an adequate approximation to biological processes.

Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic Markov processes.

Single-cell experiments show that gene expression is stochastic and bursty, a feature that can emerge from slow switching between promoter states with different activities. In addition to slow chromatin and/or DNA looping dynamics, one source of long-lived promoter states is the slow binding and unbinding kinetics of transcription factors to promoters, i.e. the non-adiabatic binding regime. Here, we introduce a simple analytical framework, known as a piecewise deterministic Markov process (PDMP), that accurately describes the stochastic dynamics of gene expression in the non-adiabatic regime. We illustrate the utility of the PDMP on a non-trivial dynamical system by analysing the properties of a titration-based oscillator in the non-adiabatic limit. We first show how to transform the underlying chemical master equation into a PDMP where the slow transitions between promoter states are stochastic, but whose rates depend upon the faster deterministic dynamics of the transcription factors regulated by these promoters. We show that the PDMP accurately describes the observed periods of stochastic cycles in activator and repressor-based titration oscillators. We then generalize our PDMP analysis to more complicated versions of titration-based oscillators to explain how multiple binding sites lengthen the period and improve coherence. Last, we show how noise-induced oscillation previously observed in a titration-based oscillator arises from non-adiabatic and discrete binding events at the promoter site.

A systematic study of the determinants of protein abundance memory in cell lineage.

Proteins are essential players of life activities. Intracellular protein levels directly affect cellular functions and cell fate. Upon cell division, the proteins in the mother cell are inherited by the daughters. However, what factors and by how much they affect this epigenetic inheritance of protein abundance remains unclear. Using both computational and experimental approaches, we systematically investigated this problem. We derived an analytical expression for the dependence of protein inheritance on various factors and showed that it agreed with numerical simulations of protein production and experimental results. Our work provides a framework for quantitative studies of protein inheritance and for the potential application of protein memory manipulation.

Predicting Gene Expression Noise from Gene Expression Variations.

The level of gene expression is known to vary from cell to cell and even in the same cell over time. This variability provides cells with the ability to mitigate environmental stresses and genetic perturbations, and facilitates gene expression evolution. Recently, many valuable gene expression noise data measured at the single-cell level and gene expression variation measured for cell populations have become available. In this chapter, we show how to perform integrative analysis using these data. Specifically, we introduce how to apply a machine learning technique (support vector regression) to explore the relationship between gene expression variations and stochastic noise.

Noise control for molecular computing.

Synthetic biology is a growing interdisciplinary field, with far-reaching applications, which aims to design biochemical systems that behave in a desired manner. With the advancement in nucleic-acid-based technology in general, and strand-displacement DNA computing in particular, a large class of abstract biochemical networks may be physically realized using nucleic acids. Methods for systematic design of the abstract systems with prescribed behaviours have been predominantly developed at the (less-detailed) deterministic level. However, stochastic effects, neglected at the deterministic level, are increasingly found to play an important role in biochemistry. In such circumstances, methods for controlling the intrinsic noise in the system are necessary for a successful network design at the (more-detailed) stochastic level. To bridge the gap, the noise-control algorithm for designing biochemical networks is developed in this paper. The algorithm structurally modifies any given reaction network under mass-action kinetics, in such a way that (i) controllable state-dependent noise is introduced into the stochastic dynamics, while (ii) the deterministic dynamics are preserved. The capabilities of the algorithm are demonstrated on a production-decay reaction system, and on an exotic system displaying bistability. For the production-decay system, it is shown that the algorithm may be used to redesign the network to achieve noise-induced multistability. For the exotic system, the algorithm is used to redesign the network to control the stochastic switching, and achieve noise-induced oscillations.

Noise Induces the Population-Level Entrainment of Incoherent, Uncoupled Intracellular Oscillators.

Intracellular oscillators entrain to periodic signals by adjusting their phase and frequency. However, the low copy numbers of key molecular players make the dynamics of these oscillators intrinsically noisy, disrupting their oscillatory activity and entrainment response. Here, we use a combination of computational methods and experimental observations to reveal a functional distinction between the entrainment of individual oscillators (e.g., inside cells) and the entrainment of populations of oscillators (e.g., across tissues). We demonstrate that, in the presence of intracellular noise, weak periodic cues robustly entrain the population averaged response, even while individual oscillators remain un-entrained. We mathematically elucidate this phenomenon, which we call stochastic population entrainment, and show that it naturally arises due to interactions between intrinsic noise and nonlinear oscillatory dynamics. Our findings suggest that robust tissue-level oscillations can be achieved by a simple mechanism that utilizes intrinsic biochemical noise, even in the absence of biochemical couplings between cells.

Information processing by simple molecular motifs and susceptibility to noise.

Biological organisms rely on their ability to sense and respond appropriately to their environment. The molecular mechanisms that facilitate these essential processes are however subject to a range of random effects and stochastic processes, which jointly affect the reliability of information transmission between receptors and, for example, the physiological downstream response. Information is mathematically defined in terms of the entropy; and the extent of information flowing across an information channel or signalling system is typically measured by the 'mutual information', or the reduction in the uncertainty about the output once the input signal is known. Here, we quantify how extrinsic and intrinsic noise affects the transmission of simple signals along simple motifs of molecular interaction networks. Even for very simple systems, the effects of the different sources of variability alone and in combination can give rise to bewildering complexity. In particular, extrinsic variability is apt to generate 'apparent' information that can, in extreme cases, mask the actual information that for a single system would flow between the different molecular components making up cellular signalling pathways. We show how this artificial inflation in apparent information arises and how the effects of different types of noise alone and in combination can be understood.

A frequency-dependent decoding mechanism for axonal length sensing.

We have recently developed a mathematical model of axonal length sensing in which a system of delay differential equations describe a chemical signaling network. We showed that chemical oscillations emerge due to delayed negative feedback via a Hopf bifurcation, resulting in a frequency that is a monotonically decreasing function of axonal length. In this paper, we explore how frequency-encoding of axonal length can be decoded by a frequency-modulated gene network. If the protein output were thresholded, then this could provide a mechanism for axonal length control. We analyze the robustness of such a mechanism in the presence of intrinsic noise due to finite copy numbers within the gene network.

Stochastic models of cellular circadian rhythms in plants help to understand the impact of noise on robustness and clock structure.

Rhythmic behavior is essential for plants; for example, daily (circadian) rhythms control photosynthesis and seasonal rhythms regulate their life cycle. The core of the circadian clock is a genetic network that coordinates the expression of specific clock genes in a circadian rhythm reflecting the 24-h day/night cycle. Circadian clocks exhibit stochastic noise due to the low copy numbers of clock genes and the consequent cell-to-cell variation: this intrinsic noise plays a major role in circadian clocks by inducing more robust oscillatory behavior. Another source of noise is the environment, which causes variation in temperature and light intensity: this extrinsic noise is part of the requirement for the structural complexity of clock networks. Advances in experimental techniques now permit single-cell measurements and the development of single-cell models. Here we present some modeling studies showing the importance of considering both types of noise in understanding how plants adapt to regular and irregular light variations. Stochastic models have proven useful for understanding the effect of regular variations. By contrast, the impact of irregular variations and the interaction of different noise sources are less well studied.