The impact of phage and phage resistance on microbial community dynamics.

Where there are bacteria, there will be bacteriophages. These viruses are known to be important players in shaping the wider microbial community in which they are embedded, with potential implications for human health. On the other hand, bacteria possess a range of distinct immune mechanisms that provide protection against bacteriophages, including the mutation or complete loss of the phage receptor, and CRISPR-Cas adaptive immunity. While our previous work showed how a microbial community may impact phage resistance evolution, little is known about the inverse, namely how interactions between phages and these different phage resistance mechanisms affect the wider microbial community in which they are embedded. Here, we conducted a 10-day, fully factorial evolution experiment to examine how phage impact the structure and dynamics of an artificial four-species bacterial community that includes either Pseudomonas aeruginosa wild-type or an isogenic mutant unable to evolve phage resistance through CRISPR-Cas. Additionally, we used mathematical modelling to explore the ecological interactions underlying full community behaviour, as well as to identify general principles governing the impacts of phage on community dynamics. Our results show that the microbial community structure is drastically altered by the addition of phage, with Acinetobacter baumannii becoming the dominant species and P. aeruginosa being driven nearly extinct, whereas P. aeruginosa outcompetes the other species in the absence of phage. Moreover, we find that a P. aeruginosa strain with the ability to evolve CRISPR-based resistance generally does better when in the presence of A. baumannii, but that this benefit is largely lost over time as phage is driven extinct. Finally, we show that pairwise data alone is insufficient when modelling our microbial community, both with and without phage, highlighting the importance of higher order interactions in governing multispecies dynamics in complex communities. Combined, our data clearly illustrate how phage targeting a dominant species allows for the competitive release of the strongest competitor while also contributing to community diversity maintenance and potentially preventing the reinvasion of the target species, and underline the importance of mapping community composition before therapeutically applying phage.

Beyond the Bristol book: Advances and perspectives in non-smooth dynamics and applications.

Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their smooth counterparts. The classic "Bristol book" [di Bernardo et al., Piecewise-smooth Dynamical Systems. Theory and Applications (Springer-Verlag, 2008)] contains a 2008 state-of-the-art review of major results and challenges in the study of non-smooth dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches, systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new areas of application, including neuroscience, biology, ecology, climate sciences, and engineering, to which the theory has been applied.

Qualitative changes in bifurcation structure for soft vs hard impact models of a vibro-impact energy harvester.

Vibro-impact phenomena in engineering systems, considered an adverse effect in some settings, are an intrinsic part of the mechanism in others. In energy harvesting, a vibro-impact component is often intentionally introduced to increase the power output or the system's bandwidth. The impacts can be treated as "hard" for instantaneous impacts or "soft" for compliant materials. Since both types of models exhibit complex dynamics, a comparison is non-trivial. We develop a soft impact model for a vibro-impact energy harvester, calibrating it with the relevant hard impact model for large stiffness, and systematically compare the different phenomena and dynamics in various compliant regimes. Numerical results are used in two different parametric analyses, considering the bifurcation diagrams in terms of device size and external forcing parameters. Varying the natural frequency of the membranes that form the impact boundaries, we observe shifts in the bifurcation structure that promote period-1 orbits for increased softness parameters, often generating higher power output, but also introducing parameter sensitivities for increased softness. Complementary analytical results reveal unstable periodic orbits and co-existing behaviors, potentially missed by computational methods, that can influence the bifurcation structure and in turn the energy output. A non-dimensional formulation highlights the significance of ratios of external and natural frequencies in delineating soft and hard impact scenarios parametrically. The soft impact model exhibits new symmetry breaking bifurcations related to key quantities that characterize the soft impact dynamics, such as the effective restitution coefficients, the impact phase, and the contact time interval, not captured by hard impact models.

Experimental observations on unsafe zones in milling processes.

The unsafe zone in machining is a region of the parameter space where steady-state cutting operations may switch to regenerative chatter for certain perturbations, and vice versa. In the case of milling processes, this phenomenon is related to the existence of an unstable quasi-periodic oscillation, the in-sets of which limit the basin of attraction of the stable periodic motion that corresponds to the chatter-free cutting process. The mathematical model is a system of time-periodic nonlinear delay differential equations. It is studied by means of a nonlinear extension of the semidiscretization method, which enables the estimation of the parameter ranges where the unsafe (also called bistable) zones appear. The theoretical results are checked with thorough experimental work: first, step-by-step parameter variations are adapted to identify hysteresis loops, then harmonic burst excitations are used to estimate the extents of the unsafe zones. The hysteresis loops are accurately distinguished from the dynamic bifurcation phenomenon that is related to the dynamic effect of slowly varying parameters. The experimental results confirm the existence of the bistable parameter regions. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

The influence of parametric and external noise in act-and-wait control with delayed feedback.

We apply several novel semi-analytic approaches for characterizing and calculating the effects of noise in a system with act-and-wait control. For concrete illustration, we apply these to a canonical balance model for an inverted pendulum to study the combined effect of delay and noise within the act-and-wait setting. While the act-and-wait control facilitates strong stabilization through deadbeat control, a comparison of different models with continuous vs. discrete updating of the control strategy in the active period illustrates how delays combined with the imprecise application of the control can seriously degrade the performance. We give several novel analyses of a generalized act-and-wait control strategy, allowing flexibility in the updating of the control strategy, in order to understand the sensitivities to delays and random fluctuations. In both the deterministic and stochastic settings, we give analytical and semi-analytical results that characterize and quantify the dynamics of the system. These results include the size and shape of stability regions, densities for the critical eigenvalues that capture the rate of reaching the desired stable equilibrium, and amplification factors for sustained fluctuations in the context of external noise. They also provide the dependence of these quantities on the length of the delay and the active period. In particular, we see that the combined influence of delay, parametric error, or external noise and on-off control can qualitatively change the dynamics, thus reducing the robustness of the control strategy. We also capture the dependence on how frequently the control is updated, allowing an interpolation between continuous and frequent updating. In addition to providing insights for these specific models, the methods we propose are generalizable to other settings with noise, delay, and on-off control, where analytical techniques are otherwise severely scarce.

Reduced α-stable dynamics for multiple time scale systems forced with correlated additive and multiplicative Gaussian white noise.

Stochastic averaging problems with Gaussian forcing have been the subject of numerous studies, but far less attention has been paid to problems with infinite-variance stochastic forcing, such as an α-stable noise process. It has been shown that simple linear systems driven by correlated additive and multiplicative (CAM) Gaussian noise, which emerge in the context of reduced atmosphere and ocean dynamics, have infinite variance in certain parameter regimes. In this study, we consider the stochastic averaging of systems where a linear CAM noise process in the infinite variance parameter regime drives a comparatively slow process. We use (semi)-analytical approximations combined with numerical illustrations to compare the averaged process to one that is forced by a white α-stable process, demonstrating consistent properties in the case of large time-scale separation. We identify the conditions required for the fast linear CAM process to have such an influence in driving a slower process and then derive an (effectively) equivalent fast, infinite-variance process for which an existing stochastic averaging approximation is readily applied. The results are illustrated using numerical simulations of a set of example systems.

Defining the Benefits of Antibiotic Resistance in Commensals and the Scope for Resistance Optimization.

Antibiotic resistance is a major medical and public health challenge, characterized by global increases in the prevalence of resistant strains. The conventional view is that all antibiotic resistance is problematic, even when not in pathogens. Resistance in commensal bacteria poses risks, as resistant organisms can provide a reservoir of resistance genes that can be horizontally transferred to pathogens or may themselves cause opportunistic infections in the future. While these risks are real, we propose that commensal resistance can also generate benefits during antibiotic treatment of human infection, by promoting continued ecological suppression of pathogens. To define and illustrate this alternative conceptual perspective, we use a two-species mathematical model to identify the necessary and sufficient ecological conditions for beneficial resistance. We show that the benefits are limited to species (or strain) interactions where commensals suppress pathogen growth and are maximized when commensals compete with, rather than prey on or otherwise exploit pathogens. By identifying benefits of commensal resistance, we propose that rather than strictly minimizing all resistance, resistance management may be better viewed as an optimization problem. We discuss implications in two applied contexts: bystander (nontarget) selection within commensal microbiomes and pathogen treatment given polymicrobial infections. IMPORTANCE Antibiotic resistance is commonly viewed as universally costly, regardless of which bacterial cells express resistance. Here, we derive an opposing logic, where resistance in commensal bacteria can lead to reductions in pathogen density and improved outcomes on both the patient and public health scales. We use a mathematical model of commensal-pathogen interactions to define the necessary and sufficient conditions for beneficial resistance, highlighting the importance of reciprocal ecological inhibition to maximize the benefits of resistance. More broadly, we argue that determining the benefits as well as the costs of resistances in human microbiomes can transform resistance management from a minimization to an optimization problem. We discuss applied contexts and close with a review of key resistance optimization dimensions, including the magnitude, spectrum, and mechanism of resistance.

The impact of phage and phage resistance on microbial community dynamics.

Where there are bacteria, there will be bacteriophages. These viruses are known to be important players in shaping the wider microbial community in which they are embedded, with potential implications for human health. On the other hand, bacteria possess a range of distinct immune mechanisms that provide protection against bacteriophages, including the mutation or complete loss of the phage receptor, and CRISPR-Cas adaptive immunity. Yet little is known about how interactions between phages and these different phage resistance mechanisms affect the wider microbial community in which they are embedded. Here, we conducted a 10-day, fully factorial evolution experiment to examine how phage impact the structure and dynamics of an artificial four-species bacterial community that includes either Pseudomonas aeruginosa wild type or an isogenic mutant unable to evolve phage resistance through CRISPR-Cas. Our results show that the microbial community structure is drastically altered by the addition of phage, with Acinetobacter baumannii becoming the dominant species and P. aeruginosa being driven nearly extinct, whereas P. aeruginosa outcompetes the other species in the absence of phage. Moreover, we find that a P. aeruginosa strain with the ability to evolve CRISPR-based resistance generally does better when in the presence of A. baumannii, but that this benefit is largely lost over time as phage is driven extinct. Combined, our data highlight how phage-targeting a dominant species allows for the competitive release of the strongest competitor whilst also contributing to community diversity maintenance and potentially preventing the reinvasion of the target species, and underline the importance of mapping community composition before therapeutically applying phage.

Characterizing mixed mode oscillations shaped by noise and bifurcation structure.

Many neuronal systems and models display a certain class of mixed mode oscillations (MMOs) consisting of periods of small amplitude oscillations interspersed with spikes. Various models with different underlying mechanisms have been proposed to generate this type of behavior. Stochastic versions of these models can produce similarly looking time series, often with noise-driven mechanisms different from those of the deterministic models. We present a suite of measures which, when applied to the time series, serves to distinguish models and classify routes to producing MMOs, such as noise-induced oscillations or delay bifurcation. By focusing on the subthreshold oscillations, we analyze the interspike interval density, trends in the amplitude, and a coherence measure. We develop these measures on a biophysical model for stellate cells and a phenomenological FitzHugh-Nagumo-type model and apply them on related models. The analysis highlights the influence of model parameters and resets and return mechanisms in the context of a novel approach using noise level to distinguish model types and MMO mechanisms. Ultimately, we indicate how the suite of measures can be applied to experimental time series to reveal the underlying dynamical structure, while exploiting either the intrinsic noise of the system or tunable extrinsic noise.

Direct observation of markovian behavior of the mechanical unfolding of individual proteins.

Single-molecule force-clamp spectroscopy is a valuable tool to analyze unfolding kinetics of proteins. Previous force-clamp spectroscopy experiments have demonstrated that the mechanical unfolding of ubiquitin deviates from the generally assumed Markovian behavior and involves the features of glassy dynamics. Here we use single molecule force-clamp spectroscopy to study the unfolding kinetics of a computationally designed fast-folding mutant of the small protein GB1, which shares a similar beta-grasp fold as ubiquitin. By treating the mechanical unfolding of polyproteins as the superposition of multiple identical Poisson processes, we developed a simple stochastic analysis approach to analyze the dwell time distribution of individual unfolding events in polyprotein unfolding trajectories. Our results unambiguously demonstrate that the mechanical unfolding of NuG2 fulfills all criteria of a memoryless Markovian process. This result, in contrast with the complex mechanical unfolding behaviors observed for ubiquitin, serves as a direct experimental demonstration of the Markovian behavior for the mechanical unfolding of a protein and reveals the complexity of the unfolding dynamics among structurally similar proteins. Furthermore, we extended our method into a robust and efficient pseudo-dwell-time analysis method, which allows one to make full use of all the unfolding events obtained in force-clamp experiments without categorizing the unfolding events. This method enabled us to measure the key parameters characterizing the mechanical unfolding energy landscape of NuG2 with improved precision. We anticipate that the methods demonstrated here will find broad applications in single-molecule force-clamp spectroscopy studies for a wide range of proteins.

A computational study of spike time reliability in two types of threshold dynamics.

Spike time reliability (STR) refers to the phenomenon in which repetitive applications of a frozen copy of one stochastic signal to a neuron trigger spikes with reliable timing while a constant signal fails to do so. Observed and explored in numerous experimental and theoretical studies, STR is a complex dynamic phenomenon depending on the nature of external inputs as well as intrinsic properties of a neuron. The neuron under consideration could be either quiescent or spontaneously spiking in the absence of the external stimulus. Focusing on the situation in which the unstimulated neuron is quiescent but close to a switching point to oscillations, we numerically analyze STR treating each spike occurrence as a time localized event in a model neuron. We study both the averaged properties as well as individual features of spike-evoking epochs (SEEs). The effects of interactions between spikes is minimized by selecting signals that generate spikes with relatively long interspike intervals (ISIs). Under these conditions, the frequency content of the input signal has little impact on STR. We study two distinct cases, Type I in which the f-I relation (f for frequency, I for applied current) is continuous and Type II where the f-I relation exhibits a jump. STR in the two types shows a number of similar features and differ in some others. SEEs that are capable of triggering spikes show great variety in amplitude and time profile. On average, reliable spike timing is associated with an accelerated increase in the "action" of the signal as a threshold for spike generation is approached. Here, "action" is defined as the average amount of current delivered during a fixed time interval. When individual SEEs are studied, however, their time profiles are found important for triggering more precisely timed spikes. The SEEs that have a more favorable time profile are capable of triggering spikes with higher precision even at lower action levels.

Stochastic phase dynamics and noise-induced mixed-mode oscillations in coupled oscillators.

Synaptically coupled neurons show in-phase or antiphase synchrony depending on the chemical and dynamical nature of the synapse. Deterministic theory helps predict the phase differences between two phase-locked oscillators when the coupling is weak. In the presence of noise, however, deterministic theory faces difficulty when the coexistence of multiple stable oscillatory solutions occurs. We analyze the solution structure of two coupled neuronal oscillators for parameter values between a subcritical Hopf bifurcation point and a saddle node point of the periodic branch that bifurcates from the Hopf point, where a rich variety of coexisting solutions including asymmetric localized oscillations occurs. We construct these solutions via a multiscale analysis and explore the general bifurcation scenario using the lambda-omega model. We show for both excitatory and inhibitory synapses that noise causes important changes in the phase and amplitude dynamics of such coupled neuronal oscillators when multiple oscillatory solutions coexist. Mixed-mode oscillations occur when distinct bistable solutions are randomly visited. The phase difference between the coupled oscillators in the localized solution, coexisting with in-phase or antiphase solutions, is clearly represented in the stochastic phase dynamics.

Sustained oscillations via coherence resonance in SIR.

Sustained oscillations in a stochastic SIR model are studied using a new multiple scale analysis. It captures the interaction of the deterministic and stochastic elements together with the separation of time scales inherent in the appearance of these dynamics. The nearly regular fluctuations in the infected and susceptible populations are described via an explicit construction of a stochastic amplitude equation. The agreement between the power spectral densities of the full model and the approximation verifies that coherence resonance is driving the behavior. The validity criteria for this asymptotic approximation give explicit expressions for the parameter ranges in which one expects to observe this phenomenon.

Noise-induced coherence and network oscillations in a reduced bursting model.

The dynamics of the Hindmarsh-Rose (HR) model of bursting thalamic neurons is reduced to a system of two linear differential equations that retains the subthreshold resonance properties of the HR model. Introducing a reset mechanism after a threshold crossing, we turn this system into a resonant integrate-and-fire (RIF) model. Using Monte-Carlo simulations and mathematical analysis, we examine the effects of noise and the subthreshold dynamic properties of the RIF model on the occurrence of coherence resonance (CR). Synchronized burst firing occurs in a network of such model neurons with excitatory pulse-coupling. The coherence level of the network oscillations shows a stochastic resonance-like dependence on the noise level. Stochastic analysis of the equations shows that the slow recovery from the spike-induced inhibition is crucial in determining the frequencies of the CR and the subthreshold resonance in the original HR model. In this particular type of CR, the oscillation frequency strongly depends on the intrinsic time scales but changes little with the noise intensity. We give analytical quantities to describe this CR mechanism and illustrate its influence on the emerging network oscillations. We discuss the profound physiological roles this kind of CR may have in information processing in neurons possessing a subthreshold resonant frequency and in generating synchronized network oscillations with a frequency that is determined by intrinsic properties of the neurons.