Amplitude metrics for cellular circadian bioluminescence reporters

Pharmacokinetic Model-Based Control across the Blood-Brain Barrier for Circadian Entrainment

The ability to shift circadian phase in vivo has the potential to offer substantial health benefits. However, the blood-brain barrier prevents the absorption of the majority of large and many small molecules, posing a challenge to neurological pharmaceutical development. Motivated by the presence of the circadian molecule KL001, which is capable of causing phase shifts in a circadian oscillator, we investigated the pharmacokinetics of different neurological pharmaceuticals on the dynamics of circadian phase. Specifically, we developed and validated five different transport models that describe drug concentration profiles of a circadian pharmaceutical at the brain level under oral administration and designed a nonlinear model predictive control (MPC)-based framework for phase resetting. Performance of the novel control algorithm based on the identified pharmacokinetic models was demonstrated through simulations of real-world misalignment scenarios due to jet lag. The time to achieve a complete phase reset for 11-h phase delay ranged between 48 and 72 h, while a 5-h phase advance was compensated in 30 to 60 h. This approach provides mechanistic insight into the underlying structure of the circadian oscillatory system and thus leads to a better understanding of the feasibility of therapeutic manipulations of the system.

Dual-Color Single-Cell Imaging of the Suprachiasmatic Nucleus Reveals a Circadian Role in Network Synchrony

The suprachiasmatic nucleus (SCN) acts as a master pacemaker driving circadian behavior and physiology. Although the SCN is small, it is composed of many cell types, making it difficult to study the roles of particular cells. Here we develop bioluminescent circadian reporter mice that are Cre dependent, allowing the circadian properties of genetically defined populations of cells to be studied in real time. Using a Color-Switch PER2::LUCIFERASE reporter that switches from red PER2::LUCIFERASE to green PER2::LUCIFERASE upon Cre recombination, we assess circadian rhythms in two of the major classes of peptidergic neurons in the SCN: AVP (arginine vasopressin) and VIP (vasoactive intestinal polypeptide). Surprisingly, we find that circadian function in AVP neurons, not VIP neurons, is essential for autonomous network synchrony of the SCN and stability of circadian rhythmicity.

Mathematical modeling of circadian rhythms

Circadian rhythms are endogenous ~24-hr oscillations usually entrained to daily environmental cycles of light/dark. Many biological processes and physiological functions including mammalian body temperature, the cell cycle, sleep/wake cycles, neurobehavioral performance, and a wide range of diseases including metabolic, cardiovascular, and psychiatric disorders are impacted by these rhythms. Circadian clocks are present within individual cells and at tissue and organismal levels as emergent properties from the interaction of cellular oscillators. Mathematical models of circadian rhythms have been proposed to provide a better understanding of and to predict aspects of this complex physiological system. These models can be used to: (a) manipulate the system in silico with specificity that cannot be easily achieved using in vivo and in vitro experimental methods and at lower cost, (b) resolve apparently contradictory empirical results, (c) generate hypotheses, (d) design new experiments, and (e) to design interventions for altering circadian rhythms. Mathematical models differ in structure, the underlying assumptions, the number of parameters and variables, and constraints on variables. Models representing circadian rhythms at different physiologic scales and in different species are reviewed to promote understanding of these models and facilitate their use. This article is categorized under: Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.

Pharmaceutical-based entrainment of circadian phase via nonlinear model predictive control

The widespread adoption of closed-loop control in systems biology has resulted from improvements in sensors, computing, actuation, and the discovery of alternative sites of targeted drug delivery. Most control algorithms for circadian phase resetting exploit light inputs. However, recently identified small-molecule pharmaceuticals offer advantages in terms of invasiveness and potency of actuation. Herein, we develop a systematic method to control the phase of biological oscillations motivated by the recently identified small molecule circadian pharmaceutical KL001. The model-based control architecture exploits an infinitesimal parametric phase response curve (ipPRC) that is used to predict the effect of control inputs on future phase trajectories of the oscillator. The continuous time optimal control policy is first derived for phase resetting, based on the ipPRC and Pontryagin's maximum principle. Owing to practical challenges in implementing a continuous time optimal control policy, we investigate the effect of implementing the continuous time policy in a sampled time format. Specifically, we provide bounds on the errors incurred by the physiologically tractable sampled time control law. We use these results to select directions of resetting (i.e. phase advance or delay), sampling intervals, and prediction horizons for a nonlinear model predictive control (MPC) algorithm for phase resetting. The potential of this ipPRC-informed pharmaceutical nonlinear MPC is then demonstrated in silico using real-world scenarios of jet lag or rotating shift work.

Dispersion of the intrinsic neuronal periods affects the relationship of the entrainment range to the coupling strength in the suprachiasmatic nucleus

Living beings on the Earth are subjected to and entrained (synchronized) to the natural 24-h light-dark cycle. Interestingly, they can also be entrained to an external artificial cycle of non-24-h periods. The range of these periods is called the entrainment range and it differs among species. In mammals, the entrainment range is regulated by a main clock located in the suprachiasmatic nucleus (SCN) which is composed of 10 000 neurons in the brain. Previous works have found that the entrainment range depends on the cellular coupling strength in the SCN. In particular, the entrainment range decreases with the increase of the cellular coupling strength, provided that all the neuronal oscillators are identical. However, the SCN neurons differ in the intrinsic periods that follow a normal distribution in a range from 22 to 28 h. In the present study, taking the dispersion of the intrinsic neuronal periods into account, we examined the relationship between the entrainment range and the coupling strength. Results from numerical simulations and theoretical analyses both show that the relationship is altered to be paraboliclike if the intrinsic neuronal periods are nonidentical, and the maximal entrainment range is obtained with a suitable coupling strength. Our results shed light on the role of the cellular coupling in the entrainment ability of the SCN network.

A systems theoretic approach to analysis and control of mammalian circadian dynamics

The mammalian circadian clock is a complex multi-scale, multivariable biological control system. In the past two decades, methods from systems engineering have led to numerous insights into the architecture and functionality of this system. In this review, we examine the mammalian circadian system through a process systems lens. We present a mathematical framework for examining the cellular circadian oscillator, and show recent extensions for understanding population-scale dynamics. We provide an overview of the routes by which the circadian system can be systemically manipulated, and present in silico proof of concept results for phase resetting of the clock via model predictive control.

GillesPy: A Python Package for Stochastic Model Building and Simulation

GillesPy is an open-source Python package for model construction and simulation of stochastic biochemical systems. GillesPy consists of a Python framework for model building and an interface to the StochKit2 suite of efficient simulation algorithms based on the Gillespie stochastic simulation algorithms (SSA). To enable intuitive model construction and seamless integration into the scientific Python stack, we present an easy to understand, action-oriented programming interface. Here, we describe the components of this package and provide a detailed example relevant to the computational biology community.

Functional network inference of the suprachiasmatic nucleus

In the mammalian suprachiasmatic nucleus (SCN), noisy cellular oscillators communicate within a neuronal network to generate precise system-wide circadian rhythms. Although the intracellular genetic oscillator and intercellular biochemical coupling mechanisms have been examined previously, the network topology driving synchronization of the SCN has not been elucidated. This network has been particularly challenging to probe, due to its oscillatory components and slow coupling timescale. In this work, we investigated the SCN network at a single-cell resolution through a chemically induced desynchronization. We then inferred functional connections in the SCN by applying the maximal information coefficient statistic to bioluminescence reporter data from individual neurons while they resynchronized their circadian cycling. Our results demonstrate that the functional network of circadian cells associated with resynchronization has small-world characteristics, with a node degree distribution that is exponential. We show that hubs of this small-world network are preferentially located in the central SCN, with sparsely connected shells surrounding these cores. Finally, we used two computational models of circadian neurons to validate our predictions of network structure.

Quantifying Stochastic Noise in Cultured Circadian Reporter Cells

Stochastic noise at the cellular level has been shown to play a fundamental role in circadian oscillations, influencing how groups of cells entrain to external cues and likely serving as the mechanism by which cell-autonomous rhythms are generated. Despite this importance, few studies have investigated how clock perturbations affect stochastic noise—even as increasing numbers of high-throughput screens categorize how gene knockdowns or small molecules can change clock period and amplitude. This absence is likely due to the difficulty associated with measuring cell-autonomous stochastic noise directly, which currently requires the careful collection and processing of single-cell data. In this study, we show that the damping rate of population-level bioluminescence recordings can serve as an accurate measure of overall stochastic noise, and one that can be applied to future and existing high-throughput circadian screens. Using cell-autonomous fibroblast data, we first show directly that higher noise at the single-cell results in faster damping at the population level. Next, we show that the damping rate of cultured cells can be changed in a dose-dependent fashion by small molecule modulators, and confirm that such a change can be explained by single-cell noise using a mathematical model. We further demonstrate the insights that can be gained by applying our method to a genome-wide siRNA screen, revealing that stochastic noise is altered independently from period, amplitude, and phase. Finally, we hypothesize that the unperturbed clock is highly optimized for robust rhythms, as very few gene perturbations are capable of simultaneously increasing amplitude and lowering stochastic noise. Ultimately, this study demonstrates the importance of considering the effect of circadian perturbations on stochastic noise, particularly with regard to the development of small-molecule circadian therapeutics.

As most organisms exist in an environment that changes predictably with a 24-hour period, highly optimized genetic circuits turn on and off the production of key regulatory proteins to anticipate the day/night cycle. In humans, the demands of a modern society have required that we deviate from this evolutionarily prescribed sleep and feeding schedule, resulting in increased long-term risks of metabolic disease. There is therefore a desire to find pharmacological treatments that would restore the normal functioning of our circadian clock despite irregular behavioral schedules. One aspect of these treatments that is often overlooked in searching for candidate drugs is how these treatments might affect the accuracy of the circadian timing system. Recording the time of each cell is possible but difficult; as a result single-cell approaches cannot be scaled up to high-throughput searches. In this paper, we show that it is possible to estimate how much the noise of a system has changed by looking only at the averaged protein production of an entire population of cells. Such an approach allows us to analyze prior data from high-throughput screens, and show that the natural clock has been highly optimized to be both accurate and high amplitude.