Overshoot in biological systems modelled by Markov chains: a non-equilibrium dynamic phenomenon

What can we learn when fitting a simple telegraph model to a complex gene expression model?

In experiments, the distributions of mRNA or protein numbers in single cells are often fitted to the random telegraph model which includes synthesis and decay of mRNA or protein, and switching of the gene between active and inactive states. While commonly used, this model does not describe how fluctuations are influenced by crucial biological mechanisms such as feedback regulation, non-exponential gene inactivation durations, and multiple gene activation pathways. Here we investigate the dynamical properties of four relatively complex gene expression models by fitting their steady-state mRNA or protein number distributions to the simple telegraph model. We show that despite the underlying complex biological mechanisms, the telegraph model with three effective parameters can accurately capture the steady-state gene product distributions, as well as the conditional distributions in the active gene state, of the complex models. Some effective parameters are reliable and can reflect realistic dynamic behaviors of the complex models, while others may deviate significantly from their real values in the complex models. The effective parameters can also be applied to characterize the capability for a complex model to exhibit multimodality. Using additional information such as single-cell data at multiple time points, we provide an effective method of distinguishing the complex models from the telegraph model. Furthermore, using measurements under varying experimental conditions, we show that fitting the mRNA or protein number distributions to the telegraph model may even reveal the underlying gene regulation mechanisms of the complex models. The effectiveness of these methods is confirmed by analysis of single-cell data for E. coli and mammalian cells. All these results are robust with respect to cooperative transcriptional regulation and extrinsic noise. In particular, we find that faster relaxation speed to the steady state results in more precise parameter inference under large extrinsic noise.

Governing principles of transcriptional logic out of equilibrium

To survive, adapt, and develop, cells respond to external and internal stimuli by tightly regulating transcription. Transcriptional regulation involves the combinatorial binding of a repertoire of transcription factors to DNA, which often results in switch-like binary outputs akin to Boolean logic gates. Recent experimental studies have demonstrated that in eukaryotes, transcription factor binding to DNA often involves energy expenditure, thereby driving the system out of equilibrium. The governing principles of transcriptional logic operations out of equilibrium remain unexplored. Here, we employ a simple two-input, single-locus model of transcription that can accommodate both equilibrium and nonequilibrium mechanisms. Using this model, we find that nonequilibrium regimes can give rise to all the logic operations accessible in equilibrium. Strikingly, energy expenditure alters the regulatory function of the two transcription factors in a mutually exclusive manner. This allows for the emergence of new logic operations that are inaccessible in equilibrium. Overall, our results show that energy expenditure can expand the range of cellular decision-making without the need for more complex promoter architectures.

Enhancement of Prebiotic Peptide Formation in Cyclic Environments

The dynamic behaviors of prebiotic reaction networks may be critically important to understanding how larger biopolymers could emerge, despite being unfavorable to form in water. We focus on understanding the dynamics of simple systems, prior to the emergence of replication mechanisms, and what role they may have played in biopolymer formation. We specifically consider the dynamics in cyclic environments using both model and experimental data. Cyclic environmental conditions prevent a system from reaching thermodynamic equilibrium, improving the chance of observing interesting kinetic behaviors. We used an approximate kinetic model to simulate the dynamics of trimetaphosphate (TP)-activated peptide formation from glycine in cyclic wet-dry conditions. The model predicts that environmental cycling allows trimer and tetramer peptides to sustain concentrations above the predicted fixed points of the model due to overshoot, a dynamic phenomenon. Our experiments demonstrate that oscillatory environments can shift product distributions in favor of longer peptides. However, experimental validation of certain behaviors in the kinetic model is challenging, considering that open systems with cyclic environmental conditions break many of the common assumptions in classical chemical kinetics. Overall, our results suggest that the dynamics of simple peptide reaction networks in cyclic environments may have been important for the formation of longer polymers on the early Earth. Similar phenomena may have also contributed to the emergence of reaction networks with product distributions determined not by thermodynamics, but rather by kinetics.

Investing in monarch conservation: understanding private funding dynamics

Non-profit environmental organizations (NGOs) rely heavily on external donors to fulfill their mandates. However, forecasting donations for long-term planning is an elusive task at best. The non-compulsory nature of donation requires NGOs to understand how donors’ attention and funding allocations change over time as conservation scenarios change and incorporate these insights into their budgeting plans. We hypothesize that an NGO can hinder its capacity to reach its conservation goals by neglecting its donor-NGO-natural system (DNNS), which is reactive to the socio-ecological context. To test our hypothesis, we compared the ecological outcomes derived from a budgeting strategy assuming donors have a fixed willingness to pay throughout the program (open-loop) against the reality that donor preferences change over time (closed-loop) based on the evolving ecological context, partly driven by the program’s actions. Our analysis was performed using two different willingness to pay (WTP) behavioural models, one representing donors informed about the success of the program supported (GPI), and another without such information (GPI), evidencing how the underlying assumptions about the target donors can radically change the organization’s fundraising strategy. Next, we used our closed-loop approach to estimate NGO’s optimal yearly donation requests to achieve a conservation target. Finally, we tested the consequences of presuming an incorrect WTP behavioural model while estimating optimal yearly donation requests by applying the optimization results from the previous step into a model parameterized with a different behavioural model. Our model was created by coupling a discrete choice experiment (DCE) and a systems dynamics model, developing a coupled social-ecological model of the eastern Monarch butterfly (Danaus plexippus), a charismatic long-distant migrant butterfly that has dwindled in numbers across North America mainly due to the increases in GMO agriculture. Our results showed a significant difference in donations received and ecological outcome forecasted by an open-loop model and the actual numbers obtained by the more real-life, closed-loop model, highlighting the importance of accounting for human behaviour during the planning phase of a long-term conservation strategy. Next, when we used our closed-loop to estimate optimal donation requests, the conservation objectives and funds raised were consistently and efficiently achieved, regardless of the underlying behavioural WTP model. We also designed novel visual tools from the behaviour WTP model exploration to bridge the gap between science insights obtained from DCEs and decision-making. However, when we used closed-loop optimal donation requests obtained from one WTP behaviour model into a simulation parameterized with different WTP behavioural models, considerable ecological and financial targets deviations arose. These deviations highlight the importance of acknowledging the dynamic nature of donor’s behaviour and the need to thoroughly characterize such behaviour. Finally, we introduce a novel forecasting tool that conservation managers will have at their disposal to improve the accuracy of their budget forecasting and, ultimately, increase the program’s success rate.

Embeddability and rate identifiability of Kimura 2-parameter matrices

Deciding whether a substitution matrix is embeddable (i.e. the corresponding Markov process has a continuous-time realization) is an open problem even for [Formula: see text] matrices. We study the embedding problem and rate identifiability for the K80 model of nucleotide substitution. For these [Formula: see text] matrices, we fully characterize the set of embeddable K80 Markov matrices and the set of embeddable matrices for which rates are identifiable. In particular, we describe an open subset of embeddable matrices with non-identifiable rates. This set contains matrices with positive eigenvalues and also diagonal largest in column matrices, which might lead to consequences in parameter estimation in phylogenetics. Finally, we compute the relative volumes of embeddable K80 matrices and of embeddable matrices with identifiable rates. This study concludes the embedding problem for the more general model K81 and its submodels, which had been initiated by the last two authors in a separate work.

Nonequilibrium nature of adaptation in bacterial chemotaxis: A fluctuation-dissipation theorem approach

Adaptation is a crucial biological function possessed by many sensory systems. In this paper, we show that the fluctuation-dissipation theorem (FDT) serves as an ideal mathematical tool to study adaptation. With the aid of the nonequilibrium FDT developed by Seifert and Speck [Europhys. Lett. 89, 10007 (2010)EULEEJ0295-507510.1209/0295-5075/89/10007], we demonstrate the nonequilibrium nature of adaptation in bacterial chemotaxis. We further show that nonequilibrium is a necessary condition for adaptation even beyond the linear response regime using the spectral theory of generator matrices. In particular, the results of this paper are irrelevant to the specific functional forms of the model parameters. This suggests that the nonequilibrium nature of adaptation is a topological property, rather than a geometric property, of the underlying biochemical reaction network.

Nonequilibrium Enhances Adaptation Efficiency of Stochastic Biochemical Systems

Adaptation is a crucial biological function possessed by many sensory systems. Early work has shown that some influential equilibrium models can achieve accurate adaptation. However, recent studies indicate that there are close relationships between adaptation and nonequilibrium. In this paper, we provide an explanation of these two seemingly contradictory results based on Markov models with relatively simple networks. We show that as the nonequilibrium driving becomes stronger, the system under consideration will undergo a phase transition along a fixed direction: from non-adaptation to simple adaptation then to oscillatory adaptation, while the transition in the opposite direction is forbidden. This indicates that although adaptation may be observed in equilibrium systems, it tends to occur in systems far away from equilibrium. In addition, we find that nonequilibrium will improve the performance of adaptation by enhancing the adaptation efficiency. All these results provide a deeper insight into the connection between adaptation and nonequilibrium. Finally, we use a more complicated network model of bacterial chemotaxis to validate the main results of this paper.

Mapping the functional versatility and fragility of Ras GTPase signaling circuits through in vitro network reconstitution

The Ras-superfamily GTPases are central controllers of cell proliferation and morphology. Ras signaling is mediated by a system of interacting molecules: upstream enzymes (GEF/GAP) regulate Ras's ability to recruit multiple competing downstream effectors. We developed a multiplexed, multi-turnover assay for measuring the dynamic signaling behavior of in vitro reconstituted H-Ras signaling systems. By including both upstream regulators and downstream effectors, we can systematically map how different network configurations shape the dynamic system response. The concentration and identity of both upstream and downstream signaling components strongly impacted the timing, duration, shape, and amplitude of effector outputs. The distorted output of oncogenic alleles of Ras was highly dependent on the balance of positive (GAP) and negative (GEF) regulators in the system. We found that different effectors interpreted the same inputs with distinct output dynamics, enabling a Ras system to encode multiple unique temporal outputs in response to a single input. We also found that different Ras-to-GEF positive feedback mechanisms could reshape output dynamics in distinct ways, such as signal amplification or overshoot minimization. Mapping of the space of output behaviors accessible to Ras provides a design manual for programming Ras circuits, and reveals how these systems are readily adapted to produce an array of dynamic signaling behaviors. Nonetheless, this versatility comes with a trade-off of fragility, as there exist numerous paths to altered signaling behaviors that could cause disease.

The overshoot and phenotypic equilibrium in characterizing cancer dynamics of reversible phenotypic plasticity

The paradigm of phenotypic plasticity indicates reversible relations of different cancer cell phenotypes, which extends the cellular hierarchy proposed by the classical cancer stem cell (CSC) theory. Since it is still questionable if the phenotypic plasticity is a crucial improvement to the hierarchical model or just a minor extension to it, it is worthwhile to explore the dynamic behavior characterizing the reversible phenotypic plasticity. In this study we compare the hierarchical model and the reversible model in predicting the cell-state dynamics observed in biological experiments. Our results show that the hierarchical model shows significant disadvantages over the reversible model in describing both long-term stability (phenotypic equilibrium) and short-term transient dynamics (overshoot) in cancer cell populations. In a very specific case in which the total growth of population due to each cell type is identical, the hierarchical model predicts neither phenotypic equilibrium nor overshoot, whereas the reversible model succeeds in predicting both of them. Even though the performance of the hierarchical model can be improved by relaxing the specific assumption, its prediction to the phenotypic equilibrium strongly depends on a precondition that may be unrealistic in biological experiments. Moreover, it still does not show as rich dynamics as the reversible model in capturing the overshoots of both CSCs and non-CSCs. By comparison, it is more likely for the reversible model to correctly predict the stability of the phenotypic mixture and various types of overshoot behavior.