A mathematical study of the efficacy of possible negative feedback pathways involved in neuronal polarization.

Neuronal polarization, a process wherein nascent neurons develop a single long axon and multiple short dendrites, can occur within in vitro cell cultures without environmental cues. This is an apparently random process in which one of several short processes, called neurites, grows to become long, while the others remain short. In this study, we propose a minimum model for neurite growth, which involves bistability and random excitations reflecting actin waves. Positive feedback is needed to produce the bistability, while negative feedback is required to ensure that no more than one neurite wins the winner-takes-all contest. By applying the negative feedback to different aspects of the neurite growth process, we demonstrate that targeting the negative feedback to the excitation amplitude results in the most persistent polarization. Also, we demonstrate that there are optimal ranges of values for the neurite count, and for the excitation rate and amplitude that best maintain the polarization. Finally, we show that a previously published model for neuronal polarization based on competition for limited resources shares key features with our best-performing minimal model: bistability and negative feedback targeted to the size of random excitations.

Hormonal regulation of ovarian follicle growth in humans: Model-based exploration of cycle variability and parameter sensitivities.

We present a modelling and simulation framework for the dynamics of ovarian follicles and key hormones along the hypothalamic-pituitary-gonadal axis throughout consecutive human menstrual cycles. All simulation results (hormone concentrations and ovarian follicle sizes) are in biological units and can easily be compared to clinical data. The model takes into account variability in follicles' response to stimulating hormones, which introduces variability between cycles. The growth of ovarian follicles in waves is an emergent property in our model simulations and further supports the hypothesis that follicular waves are also present in humans. We use Approximate Bayesian Computation and cluster analysis to construct a population of virtual subjects and to study parameter distributions and sensitivities. The model can be used to compare and optimize treatment protocols for ovarian hyperstimulation, thus potentially forming the integral part of a clinical decision support system in reproductive endocrinology.

Locomotion: exploiting noise for state estimation.

Running, walking, flying and swimming are all processes in which animals produce propulsion by executing rhythmic motions of their bodies. Dynamical stability of the locomotion is hardly automatic: millions of older people are injured by falling each year. Stability frequently requires sensory feedback. We investigate how organisms obtain the information they use in maintaining their stability. Assessing stability of a periodic orbit of a dynamical system requires information about the dynamics of the system off the orbit. For locomotion driven by a periodic orbit, perturbations that "kick" the trajectory off the orbit must occur in order to observe convergence rates toward the orbit. We propose that organisms generate excitations in order to set the gains for stabilizing feedback. We hypothesize further that these excitations are stochastic but have heavy-tailed, non-Gaussian probability distributions. Compared to Gaussian distributions, we argue that these are more effective for estimating stability characteristics of the orbit. Finally, we propose experiments to test the efficacy of these ideas.

Qualitative models and experimental investigation of chaotic NOR gates and set/reset flip-flops.

It has been observed through experiments and SPICE simulations that logical circuits based upon Chua's circuit exhibit complex dynamical behaviour. This behaviour can be used to design analogues of more complex logic families and some properties can be exploited for electronics applications. Some of these circuits have been modelled as systems of ordinary differential equations. However, as the number of components in newer circuits increases so does the complexity. This renders continuous dynamical systems models impractical and necessitates new modelling techniques. In recent years, some discrete dynamical models have been developed using various simplifying assumptions. To create a robust modelling framework for chaotic logical circuits, we developed both deterministic and stochastic discrete dynamical models, which exploit the natural recurrence behaviour, for two chaotic NOR gates and a chaotic set/reset flip-flop. This work presents a complete applied mathematical investigation of logical circuits. Experiments on our own designs of the above circuits are modelled and the models are rigorously analysed and simulated showing surprisingly close qualitative agreement with the experiments. Furthermore, the models are designed to accommodate dynamics of similarly designed circuits. This will allow researchers to develop ever more complex chaotic logical circuits with a simple modelling framework.

JacLy: a Jacobian-based method for the inference of metabolic interactions from the covariance of steady-state metabolome data.

Reverse engineering metabolome data to infer metabolic interactions is a challenging research topic. Here we introduce JacLy, a Jacobian-based method to infer metabolic interactions of small networks (<20 metabolites) from the covariance of steady-state metabolome data. The approach was applied to two different in silico small-scale metabolome datasets. The power of JacLy lies on the use of steady-state metabolome data to predict the Jacobian matrix of the system, which is a source of information on structure and dynamic characteristics of the system. Besides its advantage of inferring directed interactions, its superiority over correlation-based network inference was especially clear in terms of the required number of replicates and the effect of the use of priori knowledge in the inference. Additionally, we showed the use of standard deviation of the replicate data as a suitable approximation for the magnitudes of metabolite fluctuations inherent in the system.

Endogenous Molecular-Cellular Network Cancer Theory: A Systems Biology Approach.

In light of ever apparent limitation of the current dominant cancer mutation theory, a quantitative hypothesis for cancer genesis and progression, endogenous molecular-cellular network hypothesis has been proposed from the systems biology perspective, now for more than 10 years. It was intended to include both the genetic and epigenetic causes to understand cancer. Its development enters the stage of meaningful interaction with experimental and clinical data and the limitation of the traditional cancer mutation theory becomes more evident. Under this endogenous network hypothesis, we established a core working network of hepatocellular carcinoma (HCC) according to the hypothesis and quantified the working network by a nonlinear dynamical system. We showed that the two stable states of the working network reproduce the main known features of normal liver and HCC at both the modular and molecular levels. Using endogenous network hypothesis and validated working network, we explored genetic mutation pattern in cancer and potential strategies to cure or relieve HCC from a totally new perspective. Patterns of genetic mutations have been traditionally analyzed by posteriori statistical association approaches in light of traditional cancer mutation theory. One may wonder the possibility of a priori determination of any mutation regularity. Here, we found that based on the endogenous network theory the features of genetic mutations in cancers may be predicted without any prior knowledge of mutation propensities. Normal hepatocyte and cancerous hepatocyte stable states, specified by distinct patterns of expressions or activities of proteins in the network, provide means to directly identify a set of most probable genetic mutations and their effects in HCC. As the key proteins and main interactions in the network are conserved through cell types in an organism, similar mutational features may also be found in other cancers. This analysis yielded straightforward and testable predictions on an accumulated and preferred mutation spectrum in normal tissue. The validation of predicted cancer state mutation patterns demonstrates the usefulness and potential of a causal dynamical framework to understand and predict genetic mutations in cancer. We also obtained the following implication related to HCC therapy, (1) specific positive feedback loops are responsible for the maintenance of normal liver and HCC; (2) inhibiting proliferation and inflammation-related positive feedback loops, and simultaneously inducing liver-specific positive feedback loop is predicated as the potential strategy to cure or relieve HCC; (3) the genesis and regression of HCC is asymmetric. In light of the characteristic property of the nonlinear dynamical system, we demonstrate that positive feedback loops must be existed as a simple and general molecular basis for the maintenance of phenotypes such as normal liver and HCC, and regulating the positive feedback loops directly or indirectly provides potential strategies to cure or relieve HCC.

Efficient search and responsiveness trade-offs in a Markov chain model of evolution in dynamic environments.

Motivated by the desire to study evolutionary responsiveness in fluctuating environments, and by the current interest in analyses of evolution that merge notions of fitness maximization with dynamical systems concepts such as Lyapunov functions, this paper models natural evolution with a simple stochastic dynamical system that can be represented as a Markov chain. The process maximizes fitness globally via search and has links to information and entropy. These links suggest that a possible rationale for evolution with the exponential fitness functions observed in nature is that of optimally-efficient search in a dynamic environment, which represents the quickest trade-off of prior information about the genotype search space for search effort savings after an environment perturbation. A Lyapunov function is also provided that relates the stochastic dynamical system model with search information, and the model shows that evolution is not gradient-based but dwells longer on more fit outcomes. The model further indicates that tuning the amount of selection trades off environment responsiveness with the time to reach fit outcomes, and that excessive selection causes a loss of responsiveness, a result that is validated by the literature and impacts efforts in directed evolution.