Bifunctional enzyme provides absolute concentration robustness in multisite covalent modification networks.

Biochemical covalent modification networks exhibit a remarkable suite of steady state and dynamical properties such as multistationarity, oscillations, ultrasensitivity and absolute concentration robustness. This paper focuses on conditions required for a network of this type to have a species with absolute concentration robustness. We find that the robustness in a substrate is endowed by its interaction with a bifunctional enzyme, which is an enzyme that has different roles when isolated versus when bound as a substrate-enzyme complex. When isolated, the bifunctional enzyme promotes production of more molecules of the robust species while when bound, the same enzyme facilitates degradation of the robust species. These dual actions produce robustness in the large class of covalent modification networks. For each network of this type, we find the network conditions for the presence of robustness, the species that has robustness, and its robustness value. The unified approach of simultaneously analyzing a large class of networks for a single property, i.e. absolute concentration robustness, reveals the underlying mechanism of the action of bifunctional enzyme while simultaneously providing a precise mathematical description of bifunctionality.

Development of the sleep-wake switch in rats during the P2-P21 early infancy period.

In early infancy, rats randomly alternate between the sleeping and waking states-from postnatal day 2-10 (P2-P10), sleep and wake bouts are both exponentially distributed with increasing means, while from P10-P21 sleep and wake bout means continue to increase, though there is a striking qualitative shift in the distribution of wake bouts from exponential to power law. The behavioral states of sleep and wakefulness correspond to the activity of sleep-active and wake-active neuronal brainstem populations, with reciprocal inhibition between the two ensuring that only one population is active at a time. The locus coeruleus (LC) forms a third component of this circuit that rises in prominence during the P10-P21 period, as experimental evidence shows that an as-of-yet undeciphered interaction of the LC with sleep-active and wake-active populations is responsible for the transformation of the wake bout distribution from exponential to power law. Interestingly, the LC undergoes remarkable physiological changes during the P10-P21 period-gap junctions within the LC are pruned and network-wide oscillatory synchrony declines and vanishes. In this work, we discuss a series of models of sleep-active, wake-active, and the LC populations, and we use these models to postulate the nature of the interaction between these three populations and how these interactions explain empirical observations of sleep and wake bout dynamics. We hypothesize a circuit in which there is reciprocal excitation between the LC and wake-active population with inhibition from the sleep-active population to the LC that suppresses the LC during sleep bouts. During the P2-P10 period, we argue that a noise-based switching mechanism between the sleep-active and wake-active populations provides a simple and natural way to account for exponential bout distributions, and that the locked oscillatory state of the LC prevents it from impacting bout distributions. From P10-P21, we use our models to postulate that, as the LC gradually shifts from a state of synchronized oscillations to a state of continuous firing, reciprocal excitation between the LC and the wake-active population is able to gradually transform the wake bout distribution from exponential to power law.

Foundations of static and dynamic absolute concentration robustness.

Absolute Concentration Robustness (ACR) was introduced by Shinar and Feinberg (Science 327:1389-1391, 2010) as robustness of equilibrium species concentration in a mass action dynamical system. Their aim was to devise a mathematical condition that will ensure robustness in the function of the biological system being modeled. The robustness of function rests on what we refer to as empirical robustness-the concentration of a species remains unvarying, when measured in the long run, across arbitrary initial conditions. Even simple examples show that the ACR notion introduced in Shinar and Feinberg (Science 327:1389-1391, 2010) (here referred to as static ACR) is neither necessary nor sufficient for empirical robustness. To make a stronger connection with empirical robustness, we define dynamic ACR, a property related to long-term, global dynamics, rather than only to equilibrium behavior. We discuss general dynamical systems with dynamic ACR properties as well as parametrized families of dynamical systems related to reaction networks. We find necessary and sufficient conditions for dynamic ACR in complex balanced reaction networks, a class of networks that is central to the theory of reaction networks.

Transition graph decomposition for complex balanced reaction networks with non-mass-action kinetics.

Reaction networks are widely used models to describe biochemical processes. Stochastic fluctuations in the counts of biological macromolecules have amplified consequences due to their small population sizes. This makes it necessary to favor stochastic, discrete population, continuous time models. The stationary distributions provide snapshots of the model behavior at the stationary regime, and as such finding their expression in terms of the model parameters is of great interest. The aim of the present paper is to describe when the stationary distributions of the original model, whose state space is potentially infinite, coincide exactly with the stationary distributions of the process truncated to finite subsets of states, up to a normalizing constant. The finite subsets of states we identify are called copies and are inspired by the modular topology of reaction network models. With such a choice we prove a novel graphical characterization of the concept of complex balancing for stochastic models of reaction networks. The results of the paper hold for the commonly used mass-action kinetics but are not restricted to it, and are in fact stated for more general setting.

Multistationarity in Cyclic Sequestration-Transmutation Networks.

We consider a natural class of reaction networks which consist of reactions where either two species can inactivate each other (i.e., sequestration), or some species can be transformed into another (i.e., transmutation), in a way that gives rise to a feedback cycle. We completely characterize the capacity of multistationarity of these networks. This is especially interesting because such networks provide simple examples of "atoms of multistationarity", i.e., minimal networks that can give rise to multiple positive steady states.

Autocatalytic recombination systems: A reaction network perspective.

Autocatalytic systems called hypercycles are very often incorporated in "origin of life" models. We investigate the dynamics of certain related models called bimolecular autocatalytic systems. In particular, we consider the dynamics corresponding to the relative populations in these networks, and show that it can be analyzed using well-chosen autonomous polynomial dynamical systems. Moreover, we use results from reaction network theory to prove persistence and permanence of several families of bimolecular autocatalytic systems called autocatalytic recombination systems.

On reaction network implementations of neural networks.

This paper is concerned with the utilization of deterministically modelled chemical reaction networks for the implementation of (feed-forward) neural networks. We develop a general mathematical framework and prove that the ordinary differential equations (ODEs) associated with certain reaction network implementations of neural networks have desirable properties including (i) existence of unique positive fixed points that are smooth in the parameters of the model (necessary for gradient descent) and (ii) fast convergence to the fixed point regardless of initial condition (necessary for efficient implementation). We do so by first making a connection between neural networks and fixed points for systems of ODEs, and then by constructing reaction networks with the correct associated set of ODEs. We demonstrate the theory by constructing a reaction network that implements a neural network with a smoothed ReLU activation function, though we also demonstrate how to generalize the construction to allow for other activation functions (each with the desirable properties listed previously). As there are multiple types of 'networks' used in this paper, we also give a careful introduction to both reaction networks and neural networks, in order to disambiguate the overlapping vocabulary in the two settings and to clearly highlight the role of each network's properties.

Detailed Balance [Formula: see text] Complex Balance [Formula: see text] Cycle Balance: A Graph-Theoretic Proof for Reaction Networks and Markov Chains.

We further clarify the relation between detailed-balanced and complex-balanced equilibria of reversible chemical reaction networks. Our results hold for arbitrary kinetics and also for boundary equilibria. Detailed balance, complex balance, "formal balance," and the new notion of "cycle balance" are all defined in terms of the underlying graph. This fact allows elementary graph-theoretic (non-algebraic) proofs of a previous result (detailed balance = complex balance + formal balance), our main result (detailed balance = complex balance + cycle balance), and a corresponding result in the setting of continuous-time Markov chains.

Deterministic stability regimes and noise-induced quasistable behavior in a pair of reciprocally inhibitory neurons.

Reciprocal inhibition is a common motif exploited by neuronal networks; an intuitive and tractable way to examine the behaviors produced by reciprocal inhibition is to consider a pair of neurons that synaptically inhibit each other and receive constant or noisy excitatory driving currents. In this work, we examine reciprocal inhibition using two models (a voltage-based and a current-based integrate-and-fire model with instantaneous or temporally structured input), and we use analytic and computational tools to examine the bifurcations that occur and study the various possible monostable, bistable, and tristable regimes that can exist; we find that, depending on system parameters (and on choice of neuron model), there can exist up to 3 distinct monostable regimes (denoted M0, M1, M2), 3 distinct bistable regimes (denoted B, B1, B2), and a single tristable regime (denoted T). We also find that synaptic inhibition exerts independent control over the two neurons - inhibition from neuron 1 to neuron 2 governs the spiking behavior of neuron 2 but has no impact on the spiking behavior of neuron 1 (and vice versa). The excitatory driving current, however, does not exhibit this property - the excitatory current to neuron 1 affects the spiking behavior of both neurons 1 and 2 (as does the excitatory current to neuron 2). Furthermore, we develop a methodology to examine the behavior of the system when the excitatory driving currents are allowed to be noisy, and we investigate the relationship between the behavior of the noisy system with the stability regime of the corresponding deterministic system.

Modeling the evolving oscillatory dynamics of the rat locus coeruleus through early infancy.

The mammalian locus coeruleus (LC) is a brainstem structure that displays extensive interconnections with numerous brain regions, and in particular plays a prominent role in the regulation of sleep and arousal. Postnatal LC development is known to drastically alter sleep-wake switching behavior through early infancy, and, in rats, exerts its most significant influence from about postnatal day 8 to postnatal day 21 (P8-P21). Physiologically, several dramatic changes are seen in LC functionality through this time period. Prior to P8, LC neurons are extensively coupled via electrical gap junctions and chemical synapses, and the entire LC network exhibits synchronized ~0.3 Hz subthreshold oscillations and spiking. From P8 to P21, the network oscillation frequency rises up to ~3 Hz (at P21) while the amplitude of the network oscillation decreases. Beyond P21, synchronized network oscillations vanish and gap junction coupling is sparse or nonexistent. In this work, we develop a large-scale, biophysically realistic model of the rat LC and we use this model to examine the changing physiology of the LC through the pivotal P8-P21 developmental period. We find that progressive gap junction pruning is sufficient to account for all of the physiological changes observed from P8 to P21.

Encoding whisker deflection velocity within the rodent barrel cortex using phase-delayed inhibition.

The primary sensory feature represented within the rodent barrel cortex is the velocity with which a whisker has been deflected. Whisker deflection velocity is encoded within the thalamus via population synchrony (higher deflection velocities entail greater synchrony among the corresponding thalamic population). Thalamic (TC) cells project to regular spiking (RS) cells within the barrel cortex, as well as to inhibitory cortical fast-spiking (FS) neurons, which in turn project to RS cells. Thus, TC spikes result in EPSPs followed, with a small time lag, by IPSPs within an RS cell, and hence the RS cell decodes TC population synchrony by employing a phase-delayed inhibition synchrony detection scheme. As whisker deflection velocity is increased, the probability that an RS cell spikes rises, while jitter in the timing of RS cell spikes remains constant. Furthermore, repeated whisker deflections with fixed velocity lead to system adaptation--TC →RS, TC →FS, and FS →RS synapses all weaken substantially, leading to a smaller probability of spiking of the RS cell and increased jitter in the timing of RS cell spikes. Interestingly, RS cell activity is better able to distinguish among different whisker deflection velocities after adaptation. In this work, we construct a biophysical model of a basic 'building block' of barrel cortex - the feedforward circuit consisting of TC cells, FS cells, and a single RS cell - and we examine the ability of the purely feedforward circuit to explain the experimental data on RS cell spiking probability, jitter, adaptation, and deflection velocity discrimination. Moreover, we study the contribution of the phase-delayed inhibition network structure to the ability of an RS cell to decode whisker deflection velocity encoded via TC population synchrony.

Switching mechanisms and bout times in a pair of reciprocally inhibitory neurons.

Within the appropriate parameter regime, a deterministic model of a pair of mutually inhibitory neurons receiving excitatory driving currents exhibits bistability-each of the two stable states corresponds to one neuron being active and the other being quiescent. The presence of noise in the driving currents results in a system that randomly switches back and forth between these two states, causing alternating bouts of spiking activity. In this work, we examine the random bout durations of the two neurons and dependence on system parameters. We find that bout durations of each neuron are exponentially distributed, with changes in system parameters altering only the mean of the distribution. Synaptic inhibition independently controls the bout durations of the two neurons-the mean bout time of a neuron is a function of efferent (or outgoing) inhibition, and is independent of afferent (or incoming) inhibition. Furthermore, we find that the mean bout time of a neuron exhibits a critical dependence on the time course (rather than amplitude) of efferent inhibition-mean bout time of a neuron grows exponentially with the time course of efferent inhibition, and the growth rate of this exponential function depends only on the excitatory driving current to that neuron (and not on any other system parameters). We discuss the relevance of our results to the regulation of sleep-wake cycling by medullary and pontine structures within the brain.

Decoding synchronized oscillations within the brain: phase-delayed inhibition provides a robust mechanism for creating a sharp synchrony filter.

The widespread presence of synchronized neuronal oscillations within the brain suggests that a mechanism must exist that is capable of decoding such activity. Two realistic designs for such a decoder include: (1) a read-out neuron with a high spike threshold, or (2) a phase-delayed inhibition network motif. Despite requiring a more elaborate network architecture, phase-delayed inhibition has been observed in multiple systems, suggesting that it may provide inherent advantages over simply imposing a high spike threshold. In this work, we use a computational and mathematical approach to investigate the efficacy of the phase-delayed inhibition motif in detecting synchronized oscillations. We show that phase-delayed inhibition is capable of creating a synchrony detector with sharp synchrony filtering properties that depend critically on the time course of inputs. Additionally, we show that phase-delayed inhibition creates a synchrony filter that is far more robust than that created by a high spike threshold.

Encoding with synchrony: phase-delayed inhibition allows for reliable and specific stimulus detection.

Synchronized oscillations are observed in a diverse array of neuronal systems, suggesting that synchrony represents a common mechanism used by the brain to encode and relay information. Coherent population activity can be deciphered by a decoder neuron with a high spike threshold or by a decoder using phase-delayed inhibition. These two mechanisms are fundamentally different - a high spike threshold detects a minimum number of synchronous input spikes (absolute synchrony), while phase-delayed inhibition requires a fixed fraction of incoming spikes to be synchronous (relative synchrony). We show that, in a system with noisy encoders where stimuli are encoded through synchrony, phase-delayed inhibition enables the creation of a decoder that can respond both reliably and specifically to a stimulus, while a high spike threshold does not.

Developmental emergence of power-law wake behavior depends upon the functional integrity of the locus coeruleus.

STUDY OBJECTIVES: Daily amounts of sleep and wakefulness are accumulated in discrete bouts that exhibit distinct statistical properties. In adult mammals, sleep bout durations follow an exponential distribution whereas wake bout durations follow a power-law distribution. In infant Norway rats, however, wake bouts initially follow an exponential distribution and only transition to a power-law distribution beginning around postnatal day 15 (P15). Here we test the hypothesis that the locus coeruleus (LC), one of several wake-active nuclei in the brainstem, contributes to this developmental transition. DESIGN: At P7, rats were injected subcutaneously with saline or DSP-4, a neurotoxin that targets noradrenergic (NA) LC terminals. Then, at P21, sleep and wakefulness during the day and night were monitored. The effectiveness of DSP-4 treatment was verified by measuring NA, dopamine (DA), and serotonin (5-HT) concentration in cortical and non-cortical tissue using high performance liquid chromatography. RESULTS: In relation to controls, subjects treated with DSP-4 exhibited significant reductions only in cortical and non-cortical NA concentration. Consistent with our hypothesis, the wake bout durations of DSP-4 subjects more closely followed an exponential distribution, whereas those of control subjects followed the expected power-law distribution. Sleep bout distributions were unaffected by DSP-4. CONCLUSIONS: These results suggest that the fundamental developmental transition in the statistical structure of wake bout durations is effected in part by changes in noradrenergic LC functioning. Considered within the domain of network theory, the hub-like connectivity of the LC may have important implications for the maintenance of network function in the face of random or targeted neural degeneration.

On immunotherapies and cancer vaccination protocols: a mathematical modelling approach.

In this paper we develop a new mathematical model of immunotherapy and cancer vaccination, focusing on the role of antigen presentation and co-stimulatory signaling pathways in cancer immunology. We investigate the effect of different cancer vaccination protocols on the well-documented phenomena of cancer dormancy and recurrence, and we provide a possible explanation of why adoptive (i.e. passive) immunotherapy protocols can sometimes actually promote tumour growth instead of inhibiting it (a phenomenon called immunostimulation), as opposed to active vaccination protocols based on tumour-antigen pulsed dendritic cells. Significantly, the results of our computational simulations suggest that elevated numbers of professional antigen presenting cells correlate well with prolonged time periods of cancer dormancy.