Parameter Estimation for the Compartmental Model

An estimation procedure is obtained for a stochastic compartmental model. Compartmental analysis assumes that a system may be divided into homogeneous components, or compartments. The main theory for the compartmental system was studied by Matis and Hartley (1971) with a discrete population in a steady state. All the transitions among the particles are considered to be stochastic in nature. An estimation procedure, Regular Best Asymptotic Normal (RBAN), discussed by Chiang (1956) is investigated for a stochastic m-compartmental system. The detailed proof of the procedure is provided here. Asymptotic properties for the estimator has been studied and computation has been carried out on our proposed nonlinear model. The downhill simplex search method, originally developed by Nelder and Mead (1965), and applied to minimize our quadratic form is inherently nonlinear in nature, thus avoiding the need to evaluate any derivative for point estimation of the parameters. The procedure applied to an experimental situation involving two compartments gives very encouraging results.     

On the identification by filtering techniques of a biologicaln-compartment model in which the transport rate parameters are assumed to be stochastic processes

In this paper biological compartmental models are considered which take into account the intrinsic randomness of the transport rate parameters and their possible variability in time. An identification procedure is presented for the estimation of the stochastic processes representing the transport rate parameters of a compartmental model from a noisy input-output experiment. The problem is formulated in terms of nonlinear filtering. A simple model is discussed for the case in which the transport rate parameters are independent of each other. The possibility of testing possible important features of the behavior of the transport rate parameters is also evidenced.     

Network thermodynamics: a simulation and modeling method based on the extension of thermodynamic thinking into the realm of highly organized systems

Network thermodynamic analysis is applied to the diffusion of a single nonelectrolyte through a system having the configuration of an epithelial membrane. The example is used to illustrate the methods of nodal, loop, and cut set analysis of the steady state as well as time dependent states of any linear network. The extension to the nonlinear case is also discussed. A discussion of the relation between network thermodynamics and compartmental analysis and other approaches using kinetics is given. Two examples of additional mathematical constraints inherent in the network thermodynamic approach are also given. These are the nonadditive terms in hybrid multiport representations of coupled systems (e.g., Kedem-Katchalsky) and the principle of detailed balance applied to systems containing cycles.     

A note on open linear systems

A theorem is given which states a necessary and sufficient condition for the specific activity to be uniform throughout an open compartmental system in the steady state.     

Theoretical study of a two-dimensional autocatalytic model for calcium dynamics at the extracellular fluid-bone interface

The temporal behaviours of the nonlinear substructure of a self-organized compartmental model of calcium metabolism were investigated. The order-two autocatalytic process included in this simple two-dimensional model is compared to some secondary nucleation mechanisms which should take place at the extracellular fluid-bone interface. The model gives rise to complex dynamic behaviours, and multistability properties, involving up to two stable periodic regimes (birhythmicity), were established in different topological configurations. The bifurcations occurring on the boundaries between regions of different qualitative behaviour have been determined. These properties are discussed in relation to the dynamical behaviour of other two-variable models, especially those including the same nonlinearity.     

A class of general stochastic compartmental systems

In this paper a general class of semi-Markov compartmental systems is studied. Two models for different input processes are analysed. Attention has been paid to the recurrence times associated with each compartment and to the distribution of the number of particles in each compartment. As an example, a three-compartment system is discussed to study the movement between three health states of patients with chronic diseases.     

A note on the thermodynamics and kinetics of open and steady state systems

Some conclusions of irreversible thermodynamics are summarized. It is shown that θ, the rate of irreversible entropy production, is not minimized in the steady state. It is also postulated that multiple steady states are possible in nonlinear kinetic systems, giving rise to situations of possible biological interest. The necessity of examining particular kinetic models is mentioned.     

Hidden oscillations in generalized linear mammaillary compartmental models

Oscillations due to complex eigenvalues, known to exist but difficult to detect, are sometimes totally hidden in the output of compartmental models, i.e., none of their modes appear in the output. An example is constructed of a class of linear compartmental models with complex eigenvalues, which have oscillating modes appearing in the output for some single-pool-input/single-pool-output (SpISpO) configurations, while for other such configurations all oscillations are totally hidden in the output. To generate the example, generalized mammillary compartmental models are defined in which a central pool exchanges with peripheral submodels called clusters, through individual connector pools, and their transfer functions are calculated corresponding to all SpISpO configurations. When such a model is repetitive, i.e., when it has identical peripheral clusters, and the input or the output is in the central pool, then it is zero-state equivalent, up to a multiplicative constant, with a reduced model having one peripheral cluster only. We analyze the visibility of an eigenvalue, i.e., whether or not the modes associated with it appear in the output, for repetitive generalized mammillary models. Sufficient conditions are given for such models to have oscillating modes appearing in the impulse response for some input/output configurations, while for other such configurations all oscillations are totally hidden, i.e., none appear in the output. A particularly interesting example is presented of a class of linear models with complex eigenvalues satisfying these conditions. This class has the structure of nonlinear models used to describe the process of protein synthesis and turnover.     

Finite fluctuations and multiple steady states far from equilibrium

The role of finite fluctuations in transitions between nonequilibrium steady states in nonlinear systems is investigated. Attention is focused on a model biochemical system for which the usual deterministic chemical kinetics predicts a far-from-equilibrium region of multiple steady states. A stochastic approach to chemical kinetics is adopted to study explicitly the effect of fluctuations around the coexisting stable states on a predicted hysteresis in the transition between those states. A numerical solution of the stochastic master equation for the system yields results which differ qualitatively from predictions of the purely macroscopic theory. Possible implications of these results are considered, and several important aspects of the computational scheme are discussed in some detail.     

An environ analysis for nonlinear compartment models

Environ analysis, an input-output analysis for models of ecological systems, has been previously formulated for linear systems. This note has a twofold purpose: first, we indicate that a variation of parameters technique can be applied, at least in principle, to computeboth input and output environs; and second, we show that this technique may be used for computation of environs in nonautonomous, nonlinear compartment models. This nonlinear theory, obtained as a direct extension of dynamical system developments, allows the traditional environ partitioning of compartmental storages and flows. An example of a nonlinear nutrient-producer-consumer system whose output environs can be computed asymptotically is presented to illustrate these concepts. This research was supported by the U.S. Environmental Protection Agency under cooperative agreement R806727030.     

Chemical Reaction Systems with Toric Steady States

Mass-action chemical reaction systems are frequently used in computational biology. The corresponding polynomial dynamical systems are often large (consisting of tens or even hundreds of ordinary differential equations) and poorly parameterized (due to noisy measurement data and a small number of data points and repetitions). Therefore, it is often difficult to establish the existence of (positive) steady states or to determine whether more complicated phenomena such as multistationarity exist. If, however, the steady state ideal of the system is a binomial ideal, then we show that these questions can be answered easily. The focus of this work is on systems with this property, and we say that such systems have toric steady states. Our main result gives sufficient conditions for a chemical reaction system to have toric steady states. Furthermore, we analyze the capacity of such a system to exhibit positive steady states and multistationarity. Examples of systems with toric steady states include weakly-reversible zero-deficiency chemical reaction systems. An important application of our work concerns the networks that describe the multisite phosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism.     

Chromokinetics of metabolic pathways.

Some methods to study and intuitively understand steady-state flows in complicated metabolic pathways are discussed. For this purpose, a suitable decomposition of complex metabolic schemes into smaller subsystems is used. These independent subsystems are then interpreted as basic colors of a chromatic coloring scheme. The mixture of these basic colors allows an intuitive picture of how a steady state in a metabolic pathway can be understood. Furthermore, actions of drugs can be more easily investigated on this basis. An anaerobic variant of pyruvate metabolism in rat liver mitochondria is presented as a simple example. This experiment allows measurement of the percentage that each basic color contributes to the steady states resulting from different experimental conditions. Possible implementations of existing algorithms and rational design of new drugs are discussed. A MATHEMATICA program, based on a new algorithm for finding all basic colors of stoichiometric networks, is included.     

Metabolic isotopomer labeling systems. Part I: global dynamic behavior

In the last few years metabolic flux analysis (MFA) using carbon labeling experiments (CLE) has become a major diagnostic tool in metabolic engineering. The mathematical centerpiece of MFA is the solution of isotopomer labeling systems (ILS). An ILS is a high-dimensional nonlinear differential equation system that describes the distribution of isotopomers over a metabolic network during a carbon labeling experiment. This contribution presents a global analysis of the dynamic behavior of general ILSs. It is proven that an ILS is globally stable under very weak conditions that are always satisfied in practice. In particular it is shown that in some sense ILSs are a nonlinear extension to the classical theory of compartmental systems. The central stability condition for compartmental systems, i.e., the non-existence of traps in linear compartmental networks, is also the major stability condition for ILSs. As an important side result of the proof, it is shown that ILSs can be transformed to a cascade of linear systems with time-dependent inhomogeneous terms. This cascade structure has considerable consequences for the development of efficient numerical algorithms for the solution of ILSs and thus for MFA.     

Cellular control models with linked positive and negative feedback and delays. II. Linear analysis and local stability

An analysis of local behavior is made of two nonlinear models which incorporate both an induction or positive feedback control mechanism and a repression or negative feedback control mechanism. The systems of differential equations with delays are linearized about their equilibria. The related characteristic equations which are exponential polynomials are studied to determine the local stability of the models. Computer studies are included to show the range of stability for different parameter values, and the biological significance is discussed briefly.     

Mapping parameter spaces of biological switches

Since the seminal 1961 paper of Monod and Jacob, mathematical models of biomolecular circuits have guided our understanding of cell regulation. Model-based exploration of the functional capabilities of any given circuit requires systematic mapping of multidimensional spaces of model parameters. Despite significant advances in computational dynamical systems approaches, this analysis remains a nontrivial task. Here, we use a nonlinear system of ordinary differential equations to model oocyte selection in Drosophila, a robust symmetry-breaking event that relies on autoregulatory localization of oocyte-specification factors. By applying an algorithmic approach that implements symbolic computation and topological methods, we enumerate all phase portraits of stable steady states in the limit when nonlinear regulatory interactions become discrete switches. Leveraging this initial exact partitioning and further using numerical exploration, we locate parameter regions that are dense in purely asymmetric steady states when the nonlinearities are not infinitely sharp, enabling systematic identification of parameter regions that correspond to robust oocyte selection. This framework can be generalized to map the full parameter spaces in a broad class of models involving biological switches.     

On compartmentalization

The notion of a compartment is discussed in terms of the Markovian process. From the stochastic matrix (the elements of which are state transition probabilities between different states of a particle of a chemical element), one may find a (generally) nonstochastic matrix; the elements of this second matrix are probabilities that, starting from some initial state, the particle will reach another seleced state (W. Feller, 1962,An Introduction to Probability Theory). Forming equivalence classes of states it can be shown that the equivalence classes based on an equivalence relation, which holds for the elements of the above-mentioned nonstochastic matrix, are essential for the notion of a compartment. From this procedure it is also obvious that a rigorous definition of a physically realizable compartment is impossible. Some conclusions on the practical use of compartmental analysis are drawn.     

On the Relationship of Steady States of Continuous and Discrete Models Arising from Biology

For many biological systems that have been modeled using continuous and discrete models, it has been shown that such models have similar dynamical properties. In this paper, we prove that this happens in more general cases. We show that under some conditions there is a bijection between the steady states of continuous and discrete models arising from biological systems. Our results also provide a novel method to analyze certain classes of nonlinear models using discrete mathematics.     

Indistinguishability and identifiability analysis of linear compartmental models.

Two compartmental model structures are said to be indistinguishable if they have the same input-output properties. In cases in which available a priori information is not sufficient to specify a unique compartmental model structure, indistinguishable model structures may have to be generated and their attributes examined for relevance. An algorithm is developed that, for a given compartmental model, investigates the complete set of models with the same number of compartments and the same input-output structure as the original model, applies geometrical rules necessary for indistinguishable models, and test models meeting the geometrical criteria for equality of transfer functions. Identifiability is also checked in the algorithm. The software consists of three programs. Program 1 determines the number of locally identifiable parameters. Program 2 applies several geometrical rules that eliminate many (generally most) of the candidate models. Program 3 checks the equality between system transfer functions of the original model and models being tested. Ranks of Jacobian matrices and submatrices and other criteria are used to check patterns of moment invariants and local identifiability. Structural controllability and structural observability are checked throughout the programs. The approach was successfully used to corroborate results from examples investigated by others.     

Neural field theory with variance dynamics

Previous neural field models have mostly been concerned with prediction of mean neural activity and with second order quantities such as its variance, but without feedback of second order quantities on the dynamics. Here the effects of feedback of the variance on the steady states and adiabatic dynamics of neural systems are calculated using linear neural field theory to estimate the neural voltage variance, then including this quantity in the total variance parameter of the nonlinear firing rate-voltage response function, and thus into determination of the fixed points and the variance itself. The general results further clarify the limits of validity of approaches with and without inclusion of variance dynamics. Specific applications show that stability against a saddle-node bifurcation is reduced in a purely cortical system, but can be either increased or decreased in the corticothalamic case, depending on the initial state. Estimates of critical variance scalings near saddle-node bifurcation are also found, including physiologically based normalizations and new scalings for mean firing rate and the position of the bifurcation.