Christian Bodenstein

Modeling the seasonal adaptation of circadian clocks by changes in the network structure of the suprachiasmatic nucleus.

The dynamics of circadian rhythms needs to be adapted to day length changes between summer and winter. It has been observed experimentally, however, that the dynamics of individual neurons of the suprachiasmatic nucleus (SCN) does not change as the seasons change. Rather, the seasonal adaptation of the circadian clock is hypothesized to be a consequence of changes in the intercellular dynamics, which leads to a phase distribution of electrical activity of SCN neurons that is narrower in winter and broader during summer. Yet to understand this complex intercellular dynamics, a more thorough understanding of the impact of the network structure formed by the SCN neurons is needed. To that effect, we propose a mathematical model for the dynamics of the SCN neuronal architecture in which the structure of the network plays a pivotal role. Using our model we show that the fraction of long-range cell-to-cell connections and the seasonal changes in the daily rhythms may be tightly related. In particular, simulations of the proposed mathematical model indicate that the fraction of long-range connections between the cells adjusts the phase distribution and consequently the length of the behavioral activity as follows: dense long-range connections during winter lead to a narrow activity phase, while rare long-range connections during summer lead to a broad activity phase. Our model is also able to account for the experimental observations indicating a larger light-induced phase-shift of the circadian clock during winter, which we show to be a consequence of higher synchronization between neurons. Our model thus provides evidence that the variations in the seasonal dynamics of circadian clocks can in part also be understood and regulated by the plasticity of the SCN network structure.

Biochemical frequency control by synchronisation of coupled repressilators: an in silico study of modules for circadian clock systems

Exploration of chronobiological systems emerges as a growing research field within bioinformatics focusing on various applications in medicine, agriculture, and material sciences. From a systems biological perspective, the question arises whether biological control systems for regulation of oscillatory signals and their technical counterparts utilise similar mechanisms. If so, modelling approaches and parameterisation adopted from building blocks can help to identify general components for frequency control in circadian clocks along with gaining insight into mechanisms of clock synchronisation to external stimuli like the daily rhythm of sunlight and darkness. Phase-locked loops could be an interesting candidate in this context. Both, biology and engineering, can benefit from a unified view resulting from systems modularisation. In a first experimental study, we analyse a model of coupled repressilators. We demonstrate its ability to synchronise clock signals in a monofrequential manner. Several oscillators initially deviate in phase difference and frequency with respect to explicit reaction and diffusion rates. Accordingly, the duration of the synchronisation process depends on dedicated reaction and diffusion parameters whose settings still lack to be sufficiently captured analytically.

Calculating activation energies for temperature compensation in circadian rhythms.

Many biological species possess a circadian clock, which helps them anticipate daily variations in the environment. In the absence of external stimuli, the rhythm persists autonomously with a period of approximately 24 h. However, single pulses of light, nutrients, chemicals or temperature can shift the clock phase. In the case of light- and temperature-cycles, this allows entrainment of the clock to cycles of exactly 24 h. Circadian clocks have the remarkable property of temperature compensation, that is, the period of the circadian rhythm remains relatively constant within a physiological range of temperatures. For several organisms, temperature-regulated processes within the circadian clock have been identified in recent years. However, how these processes contribute to temperature compensation is not fully understood. Here, we theoretically investigate temperature compensation in general oscillatory systems. It is known that every oscillator can be locally temperature compensated around a reference temperature, if reactions are appropriately balanced. A balancing is always possible if the control coefficient with respect to the oscillation period of at least one reaction in the oscillator network is positive. However, for global temperature compensation, the whole physiological temperature range is relevant. Here, we use an approach which leads to an optimization problem subject to the local balancing principle. We use this approach to analyse different circadian clock models proposed in the literature and calculate activation energies that lead to temperature compensation.

Using Jensen's inequality to explain the role of regular calcium oscillations in protein activation.

Oscillations of cytosolic Ca(2 +) are very important for cellular signalling in excitable and non-excitable cells. The information of various extracellular stimuli is encoded into oscillating patterns of Ca(2 +) that subsequently lead to the activation of different Ca(2 +)-sensitive target proteins in the cell. The question remains, however, why this information is transmitted by means of an oscillating rather than a constant signal. Here we show that, in fact, Ca(2 +) oscillations can achieve a better activation of target proteins than a comparable constant signal with the same amount of Ca(2 +) used. For this we use Jensens inequality that describes the relation between the function value of the average of a set of argument values and the average of the function values of the arguments from that set. We analyse the role of the cooperativity of the binding of Ca(2 +) and of zero-order ultrasensitivity, which are two properties that are often observed in experiments on the activation of Ca(2 +)-sensitive target proteins. Our results apply to arbitrary oscillation shapes and a very general decoding model, thus generalizing the observations of several previous studies. We compare our results with data from experimental studies investigating the activation of nuclear factor of activated T cells (NFAT) and Ras by oscillatory and constant signals. Although we are restricted to specific approximations due to the lack of detailed kinetic data, we find good qualitative agreement with our theoretical predictions.

Membrane Systems and Tools Combining Dynamical Structures with Reaction Kinetics for Applications in Chronobiology

This chapter addresses three coordinated chronobiological studies demonstrating the convergence of experimental observations, fine-grained spatio-temporal modelling, and predictive simulation. Due to the discrete manner of molecular assembly in cell signalling and gene regulation, we define a framework of membrane systems equipped with discretised forms of reaction kinetics in concert with variable intramolecular structures. Our first study is dedicated to circadian clocks inducing daily biological rhythms. As an inspiring example, the KaiABC core oscillator reaches its functionality by cyclically varying protein structures. Within our second study, we present a meta-model of an entire circadian clockwork able to adapt its oscillation to an external stimulus in terms of a frequency control system acting in a phase-locked loop. Substrate concentration courses resulting from gene expression reflect its oscillatory behaviour utilised in a periodical trigger for subsequent processes. In this context, our third study cytometrically quantifies the dynamical behaviour of a bistable toggle switch resulting from mutual gene regulation.

Chemical analog computers for clock frequency control based on p modules

Living organisms comprise astonishing capabilities of information processing for efficient adaptation to environmental changes. Resulting chemical control loops and regulator circuits are expected to exhibit a high functional similarity to technical counterparts subsumed by analog computers. A fascinating example is given by circadian clocks providing an endogenous biological rhythm adapted to the daily variation of sunlight and darkness. Its underlying biochemical principle of operation suggests a general functional scheme corresponding to frequency control using phase-locked loops (PLL). From a systems biology point of view, clock systems can be decomposed into specific modules like low-pass filters, arithmetic signal comparators, and controllable core oscillators. Each of them processes analog chemical signals on the fly. We introduce P modules in order to capture structure, behaviour, and interface of pure chemical analog computer models in terms of building blocks along with two simulation case studies. The first one is focused on chemical analog computer components including a controllable Goodwin-type core oscillator while the second one evolves an entire PLL-based frequency control by means of a pure chemical circadian clock model.

Maintenance of chronobiological information by p system mediated assembly of control units for oscillatory waveforms and frequency

Oscillatory signals turn out to be reliable carriers for efficient processing and propagation of information in both spheres, life sciences and engineering. Each living organism typically comprises a variety of inherent biological rhythms whose periodicities cover a widespread range of scales like split seconds, minutes, or hours, and sometimes even months or years. Due to different molecular principles of generation, those rhythms seem to persist independently from each other. Their combination and assembly in conjunction with recurrent environmental changes can lead to astonishing capabilities and evolutionary advantages. Motivated by the question on how populations of cicadas, an insect species living in the soil, sustain a synchronous life cycle of 17 years away from any known external stimulus of this duration, we aim at exploring potential underlying mechanisms by P system mediated assembly of a set of chemical control units. To this end, we identify a collection of core oscillators responsible for sinusoidal, spiking, and plated waveforms along with pass filters, switches, and modulators. Considering these units as genotypic elementary components, we utilise P system control for selection and (re-)assembly of units towards complex phenotypic systems. Two simulation case studies demonstrate the potential of this approach following the idea of artificial evolution. Our first study inspired by the cicadas converts a chemical frequency divider model 1:17 into counterparts of 1:3, 1:5, and 1:6 just by exchange of single units. In the second study adopted from the mammalian circadian clock system residing within the suprachiasmatic nucleus, we illustrate the stabilisation of the overall clock signal by addition of auxiliary core oscillators.

Modeling temperature entrainment of circadian clocks using the Arrhenius equation and a reconstructed model from Chlamydomonas reinhardtii

Endogenous circadian rhythms allow living organisms to anticipate daily variations in their natural environment. Temperature regulation and entrainment mechanisms of circadian clocks are still poorly understood. To better understand the molecular basis of these processes, we built a mathematical model based on experimental data examining temperature regulation of the circadian RNA-binding protein CHLAMY1 from the unicellular green alga Chlamydomonas reinhardtii, simulating the effect of temperature on the rates by applying the Arrhenius equation. Using numerical simulations, we demonstrate that our model is temperature-compensated and can be entrained to temperature cycles of various length and amplitude. The range of periods that allow entrainment of the model depends on the shape of the temperature cycles and is larger for sinusoidal compared to rectangular temperature curves. We show that the response to temperature of protein (de)phosphorylation rates play a key role in facilitating temperature entrainment of the oscillator in Chlamydomonas reinhardtii. We systematically investigated the response of our model to single temperature pulses to explain experimentally observed phase response curves.

Jensen's inequality as a tool for explaining the effect of oscillations on the average cytosolic calcium concentration

It has often been asked which physiological advantages calcium (Ca2+) oscillations in non-excitable cells may have as compared to an adjustable stationary Ca2+ signal. One of the proposed answers is that an oscillatory regime allows a lowering of the average Ca2+ concentration, which is likely to be advantageous because Ca2+ is harmful to the cell in high concentrations. To check this hypothesis, we apply Jensen’s inequality to study the relation between the average Ca2+ concentration during oscillations and the Ca2+ concentration at the (unstable) steady state. Jensen’s inequality states that for a (strictly) convex function, the function value of the average of a set of argument values is lower than the average of the function values of the arguments from that set. We show that the kinetics of the Ca2+ efflux out of the cell is crucial in this context. By analytical calculations we derive that, if the Ca2+ efflux is a convex function of the cytosolic Ca2+ concentration, then oscillations lower the average Ca2+ concentration in comparison to the unstable steady state. If it is a concave function, the average Ca2+ concentration is increased, while it remains the same if that function is linear. We also analyse the case where the efflux obeys a Hill kinetics, which involves both a convex and a concave part. The results are illustrated by numerical simulations and simple example models. The theoretical predictions are tested with three experimental data sets from the literature. In two of them, the average appears to be higher than the steady-state value, while the third points to approximate equality. Thus oscillations may be used in real cells to tune the average Ca2+ concentration in both directions.