Hybrid stochastic simulations of intracellular reaction-diffusion systems.

With the observation that stochasticity is important in biological systems, chemical kinetics have begun to receive wider interest. While the use of Monte Carlo discrete event simulations most accurately capture the variability of molecular species, they become computationally costly for complex reaction-diffusion systems with large populations of molecules. On the other hand, continuous time models are computationally efficient but they fail to capture any variability in the molecular species. In this study a hybrid stochastic approach is introduced for simulating reaction-diffusion systems. We developed an adaptive partitioning strategy in which processes with high frequency are simulated with deterministic rate-based equations, and those with low frequency using the exact stochastic algorithm of Gillespie. Therefore the stochastic behavior of cellular pathways is preserved while being able to apply it to large populations of molecules. We describe our method and demonstrate its accuracy and efficiency compared with the Gillespie algorithm for two different systems. First, a model of intracellular viral kinetics with two steady states and second, a compartmental model of the postsynaptic spine head for studying the dynamics of Ca+2 and NMDA receptors.

Extended kalman filter for estimation of parameters in nonlinear state-space models of biochemical networks.

It is system dynamics that determines the function of cells, tissues and organisms. To develop mathematical models and estimate their parameters are an essential issue for studying dynamic behaviors of biological systems which include metabolic networks, genetic regulatory networks and signal transduction pathways, under perturbation of external stimuli. In general, biological dynamic systems are partially observed. Therefore, a natural way to model dynamic biological systems is to employ nonlinear state-space equations. Although statistical methods for parameter estimation of linear models in biological dynamic systems have been developed intensively in the recent years, the estimation of both states and parameters of nonlinear dynamic systems remains a challenging task. In this report, we apply extended Kalman Filter (EKF) to the estimation of both states and parameters of nonlinear state-space models. To evaluate the performance of the EKF for parameter estimation, we apply the EKF to a simulation dataset and two real datasets: JAK-STAT signal transduction pathway and Ras/Raf/MEK/ERK signaling transduction pathways datasets. The preliminary results show that EKF can accurately estimate the parameters and predict states in nonlinear state-space equations for modeling dynamic biochemical networks.

Learning reward timing in cortex through reward dependent expression of synaptic plasticity.

The ability to represent time is an essential component of cognition but its neural basis is unknown. Although extensively studied both behaviorally and electrophysiologically, a general theoretical framework describing the elementary neural mechanisms used by the brain to learn temporal representations is lacking. It is commonly believed that the underlying cellular mechanisms reside in high order cortical regions but recent studies show sustained neural activity in primary sensory cortices that can represent the timing of expected reward. Here, we show that local cortical networks can learn temporal representations through a simple framework predicated on reward dependent expression of synaptic plasticity. We assert that temporal representations are stored in the lateral synaptic connections between neurons and demonstrate that reward-modulated plasticity is sufficient to learn these representations. We implement our model numerically to explain reward-time learning in the primary visual cortex (V1), demonstrate experimental support, and suggest additional experimentally verifiable predictions.

Interlinked dual-time feedback loops can enhance robustness to stochasticity and persistence of memory.

Multiple interlinked positive feedback loops shape the stimulus responses of various biochemical systems, such as the cell cycle or intracellular Ca2+ release. Recent studies with simplified models have identified two advantages of coupling fast and slow feedback loops. This dual-time structure enables a fast response while enhancing resistances of responses and bistability to stimulus noise. We now find that (1) the dual-time structure similarly confers resistance to internal noise due to molecule number fluctuations, and (2) model variants with altered coupling, which better represent some specific biochemical systems, share all the above advantages. We also develop a similar bistable model with coupling of a fast autoactivation loop to a slow loop. This model's topology was suggested by positive feedback proposed to play a role in long-term synaptic potentiation (LTP). The advantages of fast response and noise resistance are also present in this autoactivation model. Empirically, LTP develops resistance to reversal over approximately 1h . The model suggests this resistance may result from increased amounts of synaptic kinases involved in positive feedback.

Evaluating statistical methods used to estimate the number of postsynaptic receptors.

Calcium levels in spines play a significant role in determining the sign and magnitude of synaptic plasticity. The magnitude of calcium influx into spines is highly dependent on influx through N-methyl D-aspartate (NMDA) receptors, and therefore depends on the number of postsynaptic NMDA receptors in each spine. We have calculated previously how the number of postsynaptic NMDA receptors determines the mean and variance of calcium transients in the postsynaptic density, and how this alters the shape of plasticity curves. However, the number of postsynaptic NMDA receptors in the postsynaptic density is not well known. Anatomical methods for estimating the number of NMDA receptors produce estimates that are very different than those produced by physiological techniques. The physiological techniques are based on the statistics of synaptic transmission and it is difficult to experimentally estimate their precision. In this paper we use stochastic simulations in order to test the validity of a physiological estimation technique based on failure analysis. We find that the method is likely to underestimate the number of postsynaptic NMDA receptors, explain the source of the error, and re-derive a more precise estimation technique. We also show that the original failure analysis as well as our improved formulas are not robust to small estimation errors in key parameters.

Improving the accuracy of radiation pneumonitis dose response models

The prognosis for lung cancer patients remains poor. Five year survival rates have been reported to be 15%. Studies have shown that dose escalation to the tumor can lead to better local control and subsequently better overall survival. However, dose to lung tumor is limited by normal tissue toxicity. The most prevalent thoracic toxicity is radiation pneumonitis. In order to determine a safe dose that can be delivered to the healthy lung, researchers have turned to mathematical models predicting the rate of radiation pneumonitis. However, these models rely on simple metrics based on the dose-volume histogram and are not yet accurate enough to be used for dose escalation trials. The purpose of this work was to improve the fit of predictive risk models for radiation pneumonitis and to show the dosimetric benefit of using the models to guide patient treatment planning. The study was divided into 3 specific aims. The first two specifics aims were focused on improving the fit of the predictive model. In Specific Aim 1 we incorporated information about the spatial location of the lung dose distribution into a predictive model. In Specific Aim 2 we incorporated ventilation-based functional information into a predictive pneumonitis model. In the third specific aim a proof of principle virtual simulation was performed where a model-determined limit was used to scale the prescription dose. The data showed that for our patient cohort, the fit of the model to the data was not improved by incorporating spatial information. Although we were not able to achieve a significant improvement in model fit using pre-treatment ventilation, we show some promising results indicating that ventilation imaging can provide useful information about lung function in lung cancer patients. The virtual simulation trial demonstrated that using a personalized lung dose limit derived from a predictive model will result in a different prescription than what was achieved with the clinically used plan; thus demonstrating the utility of a normal tissue toxicity model in personalizing the prescription dose.

The volume effect in irradiated mouse colorectum

Damage of the colorectum is the dose-limiting normal tissue complication following radiotherapy of prostate and cervical cancers. One approach for decreasing complications is to physically reduce the treatment volume. Mathematical models have been previously developed to describe the change in associated toxicity with a change in irradiated volume, i.e. the "volume effect", for serial-type normal tissues including the colorectum. The first goal of this thesis was to test the hypothesis that there would not be a threshold length in the development of obstruction after irradiation of mouse colorectum, as predicted by the Probability model of the volume effect. The second goal was to examine if there were differences in the threshold and in the incidence of colorectal obstruction after irradiation of two mouse strains, C57B1/6 (C57) and C3Hf/Kam (C3H), previously found to be fibrosis-prone and-resistant, respectively, after lung irradiation due, in part, to genetic differences. The hypothesis examined was that differences in incidence between strains were due to the differential expression of the fibrogenic cytokines $\rm TGF\beta$ and $\rm TNF\alpha.$ Various lengths of C57 and C3H mouse colorectum were irradiated and the incidence of colorectal obstruction was followed up to 15 months. A threshold length was observed for both mouse strains, in contradiction of model predictions. The mechanism of the threshold was epithelial regeneration after irradiation. C57 mice had significantly higher incidence of colorectal obstruction compared to C3H mice, especially at smaller irradiated lengths. Colorectal tissue was obtained at various times after irradiation and prepared for histology, immunohistochemistry and RNase protection assay for measurement of $\rm TGF\beta 1,$ 2, 3 and $\rm TNF\alpha$ mRNA. Distinct strain differences in the histological time of appearance and spatial locations of fibrosis were observed. However, there were no consistent strain difference in mRNA levels or immunolocalization for any of the cytokines examined. The data indicate the need for volume effect models that account for biologically important processes, such as the effect of epithelial regeneration after irradiation. As well, changes in fibrogenic cytokines at the mRNA level do not contribute to the strain difference in radiation-induced colorectal obstruction. ^