Probabilistic approximations of ODEs based bio-pathway dynamics

Bio-chemical networks are often modeled as systems of ordinary differential equations (ODEs). Such systems will not admit closed form solutions and hence numerical simulations will have to be used to perform analyses. However, the number of simulations required to carry out tasks such as parameter estimation can become very large. To get around this, we propose a discrete probabilistic approximation of the ODEs dynamics. We do so by discretizing the value and the time domain and assuming a distribution of initial states w.r.t. the discretization. Then we sample a representative set of initial states according to the assumed initial distribution and generate a corresponding set of trajectories through numerical simulations. Finally, using the structure of the signaling pathway we encode these trajectories compactly as a dynamic Bayesian network.This approximation of the signaling pathway dynamics has several advantages. First, the discretized nature of the approximation helps to bridge the gap between the accuracy of the results obtained by ODE simulation and the limited precision of experimental data used for model construction and verification. Second and more importantly, many interesting pathway properties can be analyzed efficiently through standard Bayesian inference techniques instead of resorting to a large number of ODE simulations. We have tested our method on ODE models of the EGF-NGF signaling pathway [1] and the segmentation clock pathway [2]. The results are very promising in terms of accuracy and efficiency.     

A program to estimate insulin sensitivity and pancreatic responsivity from an IVGTT using the minimal modeling technique

A user-friendly program coded in PASCAL for the IBM PC has been developed to determine the etiology of impaired glucose tolerance using an intravenous glucose tolerance test (IVGTT). It makes use of the "minimal modeling technique," a method that has been shown to be adequate for the quantitative determination of insulin sensitivity and insulin resistance. Two models are used, the minimal model of glucose disappearance and the minimal model of insulin kinetics. The first model is described by two nonlinear ordinary differential equations (ODEs) which are solved numerically, and which yield the insulin sensitivity index SI. The second model is described by an ODE for which an explicit solution was obtained, and which yields the pancreatic responsivity parameters phi 1 and phi 2. The product SI.phi 2 can be used to segregate subjects into "good" and "low" tolerance types. The program provides best-fit plots along with numerical values of the parameters and their uncertainties, and requires little intervention from the user. The fact that it requires a noninvasive IVGTT as input and that it has been written for the ubiquitous IBM PC are added advantages.     

Microcomputer simulation of steady-state enzyme kinetics for educational purposes

A BASIC program to assist the instruction of steady-state enzyme kinetics has been developed for the IBM PC microcomputer. Its purpose is to simulate laboratory experiments in order to minimize the time required to obtain kinetic data from which students deduce kinetic mechanisms and determine kinetic constants of enzyme-catalyzed reactions. The program randomly selects a kinetic scheme from various sequential, ping pong, and iso reaction sequences as well as values for the kinetic constants. The scheme and kinetic constants are unknown to the student at this time; the only thing he or she knows is the stoichiometry of the catalyzed reaction which can have two or three substrates and products. The student is prompted to enter values for concentrations of substrates and products; several different concentrations for each substrate and product can be entered in a single experiment. The program then calculates, displays and prints (if desired) the corresponding initial steady-state velocities. The student can perform as many experiments as desired until enough information is obtained to determine the kinetic mechanism and to calculate values for the kinetic constants.     

Tracing pathway activities with kinase inhibitors and reverse phase protein arrays

Controlling aberrant protein kinase activity is a promising strategy for a variety of diseases, particularly cancer. Hence, the development of kinase inhibitors is currently a focal point for pharmaceutical research. In this study we utilize a chip-based reverse phase protein array (RPA) platform for profiling of kinase inhibitors in cell-based assays. In combination with the planar wave-guide technology the assay system has an absolute LOD down to the low zeptomole range. A431 cell lysates were analyzed for the activation state of key effectors in the epidermal growth factor (EGF) and insulin signaling pathways to validate this model for compound screening. A microtiter-plate format for growing, treating, and lysing cells was shown to be suitable for this approach, establishing the value of the technology as a screening tool for characterization of large numbers of kinase inhibitors against a wide variety of cellular signaling pathways. Moreover, the reverse array format allows rapid development of site-specific phosphorylation assays, since in contrast to ELISA type systems only a single antigen-specific antibody is required.     

Natural killer or T-lymphocyte cells: Which is the best immune therapeutic agent for cancer? An optimal control approach

Therapeutic vaccines are being developed as a promising new approach to treatment for cancer patients. There are still many unanswered questions about which kind of therapeutic vaccines are the best for the cancer treatments? In this paper we consider a mathematical model, in the form of a system of ordinary differential equations (ODE), this system is an example from a class of mathematical models for immunotherapy of the tumor that were derived from a biologically validated model by Lisette G. de Pillis. The problem how to schedule a variable amount of which vaccines to achieve a maximum reduction in the primary cancer volume is consider as an optimal control problem and it is shown that optimal control is quadratic with 0 denoting a trajectory corresponding to no treatment and 1 a trajectory with treatment at maximum dose along that all therapeutics are being exhausted. The ODE system dynamics characterized by locating equilibrium points and stability properties are determined by using appropriate Lyapunov functions. Especially we attend a parametric sensitivity analysis, which indicates the dependency of the optimal solution with respect to disturbances in model parameters.     

Stochastic Simulation of Ligand–Receptor Interaction

We have developed an algorithm for the stochastic simulation of ligand–receptor interactions based on 10⁴–10⁵fictitious binding sites. Reversible receptor binding was simulated by alternate random selection of sites, the first selection resulting in “occupation” if the selected site was “free,” the second selection resulting in “dissociation” if the selected site was “occupied.” We show that the mathematical formalism of mass action kinetics is predicted on purely statistical grounds. The model was extended by the introduction of two further selections, simulating a conformational change in the ligand–receptor complex (“receptor isomerization model”). All random selections were gauged separately by “probability barriers,” taking the place of macroscopic kinetic rate constants. Simulation of gradual increases and gradual decreases of the fraction of occupied fictitious binding sites in the receptor isomerization model, using various combinations of “rate constants,” resulted in biexponential time dependencies, in agreement with predictions from the integrated rate equations. Stochastic simulation of molecular processes is a powerful and versatile technique, providing the researcher with a means of studying mechanisms of increasing complexity.     

Analytical and numerical study of photocurrent transients in organic polymer solar cells

This article is an attempt to provide a self consistent picture, including existence analysis and numerical solution algorithms, of the mathematical problems arising from modeling photocurrent transients in organic polymer solar cells (OSCs). The mathematical model for OSCs consists of a system of nonlinear diffusion–reaction partial differential equations (PDEs) with electrostatic convection, coupled to a kinetic ordinary differential equation (ODE). We propose a suitable reformulation of the model that allows us to prove the existence of a solution in both stationary and transient conditions and to better highlight the role of exciton dynamics in determining the device turn-on time. For the numerical treatment of the problem, we carry out a temporal semi-discretization using an implicit adaptive method, and the resulting sequence of differential subproblems is linearized using the Newton–Raphson method with inexact evaluation of the Jacobian. Then, we use exponentially fitted finite elements for the spatial discretization, and we carry out a thorough validation of the computational model by extensively investigating the impact of the model parameters on photocurrent transient times.     

线上求解法和MATLAB用于色谱分离动力学模拟

以线性推动力模型为基础,建立考虑轴向扩散、传质阻力和非线性吸附的色谱分离动力学模型,得到一组偏微分方程,然后用线上求解法将偏微分方程转化为常微分方程,采用MATLAB的常微分方程求解器求解。部分模拟结果与试验值进行了比较,结果表明,线上求解法结合MATLAB的常微分求解器可快速、准确地模拟色谱分离过程。     

Systematic identifiability study based on the Fisher Information Matrix for reducing the number of parameters calibration of an activated sludge model

This work proposes a procedure for calibration and validation of complex models by systematically obtaining identifiable parameter subsets according to the available data. The procedure uses the new RDE criteria calculated from the Fisher Information Matrix (FIM) as the ratio of normalized D to modified E criteria (RDE). It does not require expert knowledge and it defines automatically the dimension of the identifiable subset without requiring a threshold for the RDE. It was applied successfully to the study of the IWA-ASM2d model, which was implemented, calibrated and validated for an anaerobic/anoxic/oxic (A²/O) pilot WWTP operated under three different influent ammonium concentrations (15, 20 and 30 mg/L) and two internal recycling ratios (IRR = 2 and 5). Starting from 51 among all the ASM2d parameters, a sensitivity analysis around the ASM2d default values was performed. From the sensitivity ranking, the 20 best-ranked parameters were named “seeds”, since each one served for growing a parameter subset for model calibration. The subset generation process added to the seed a parameter that presented the highest RDE among all the remaining parameters of the sensitivity ranking. The process of parameter addition was repeated until the RDE decreased from the current iteration to the previous one. The best subset determined by the methodology {b_PAO, Y_PO4, μ_A} presented the highest possible value of the RDE. Finally, the simulation of the WWTP with this subset fitted adequately the experimental data while the parameters obtained had low confidence intervals.     

A set of programs for analysis of kinetic and equilibrium data

A program package that can be used for analysis of a wide range of kinetic and equilibrium data is described. The four programs were written in Turbo Pascal and run on PC, XT, AT and compatibles. The first of the programs allows the user to fit data with 16 predefined and one user-defined function, using two different non-linear least-squares procedures. Two additional programs are used to test both the evaluation of model functions and the least-squares fits. One of these programs uses two simple procedures to generate a Gaussian-distributed random variable that is used to simulate the experimental error of measurements. The last program simulates kinetics described by differential equations that cannot be solved analytically, using numerical integration. This program helps the user to judge the validity of steady-state assumptions or treatment of kinetic measurements as relaxations.