On stochastic compartmental modeling

This communication contains a proof of the fact that the coefficient of variation of the contents of a compartment of a stochastic compartmental model with deterministic rate parameters is small for large populations. We can therefore conclude that the use of stochastic compartmental models is not of great consequence in the case of systems involving large populations when only the randomness of the transfer mechanism is considered.     

On stochastic compartmental models

The statistical averaging of compartmental models with parameters being random processes is derived for the case of vanishing input and uniformly bounded input. The difference of resulting models is discussed.     

On complex eigenvalues of compartmental models

Conditions under which the eigenvalues of the matrix of a compartmental model have a nonzero imaginary part are studied. Inequalities for the total imaginary part are obtained. The effect of excretions and combinations of cycles on this imaginary part are studied.     

On washout in nonlinear compartmental systems

Let x(t) be a solution of a compartmental system. If, for some compartment j, x_j(t)→0 as t→∞, then we say that the compartment j washes out. We show that a compartment washes out if it always reaches (along a fixed path) either the environment or another compartment for which there is no return path. Additional criteria, particularly regarding exponential convergence, are also presented. Examples are drawn from tracer kinetics, enzyme reactions, and epidemic models.     

On some topological properties of a strongly connected compartmental system with application to the identifiability problem

In this paper some structural properties of a strongly connected compartmental system are illustrated. In particular a suitable set of “cycles” and “paths” associated to the compartmental graph is constructed, such that an application exists between the parameter space and the space of sycles and paths, whose suitable restriction is a bijection. It is shown that this set contains the minimum number of functions necessary to uniquely identify the parametrization vector, and its relevance in identifiability analysis is illustrated.     

On the visibility of leading eigenvalues in compartmental models

The system matrix eigenvalue with the largest real part (leading eigenvalue) of any input-output (IO) connectable compartmental model is real and visible (appears explicitly) in the impulse response, and thus it governs the asymptotic response of the model. Its visible multiplicity is calculated here by decomposing the model into strongly connected components and applying the Perron-Frobenius theorem. Cascade models and fully visible eigenvalues are defined, and it is shown that for any cascade model the leading eigenvalue is fully visible in the impulse response. A necessary and sufficient condition is given for full visibility of the leading eigenvalue of any IO-connectable model. As a corollary, if an IO-connectable compartmental model has one or more traps, the leading eigenvalue λ₁ = 0 always has visible multiplicity one.     

On linear stochastic compartmental models in discrete time

This note presents a general time-dependent study of linear stochastic compartmental models in discrete time. The transient distribution of the state of the system is obtained by adapting methods used in the continuous time analysis. Covariance functions with and without a time lag are then deduced by a simple probabilistic argument. Results are derived in the Markov case and are partly extended to the semi-Markov case.     

Stochastic analysis of environmental fluctuations in a compartmental system

In this paper the general two-compartment system with environmental stochasticity is investigated. The transfer rates and outputs are assumed to be random telegraph processes. The Laplace-transform of the mean-value function of the amount of substance present in the two compartments is evaluated. The Gaussian white noise limit case is also discussed and the stability of the system is examined. It is shown that while the deterministic model is stable, environmental stochasticity may induce in the mean-value function all sorts of behaviour — stable, unstable and oscillatory. This is in contrast to the intrinsic stochasticity in linear models wherefor the mean-value function is the same as the solution of the deterministic counterpart.     

On the discrimination between symmetric pentapeptides

Patchornik has pointed out that identification of the dipeptides AC, GC, GA, and AG is insufficient information to allow one to distinguish between the pentapeptide sequences ACGAG and AGACG. The present study proposes a probabilistic approach which will sometimes allow one to distinguish between the parent pentapeptides when all the dipeptides have been determined quantitatively. For the case when the free amino acids have also been quantitatively determined, necessary and sufficient conditions are established for unambiguous discrimination.     

On the synthesis of viral ribonucleic acids and ribonucleoproteins in the submitochondrial system completely free of interfering cytoplasmic contaminations

Summary The synthesis of virus-specific macromolecules was studied in the reconstituted system containing inner membrane-matrix fraction from rat liver mitochondria and infectious RNA of Venezuelian equine encephalomyelitis (VEE) virus. In a series of preliminary experiments it was shown that isolated submitochondrial fraction was completely free of interfering cytoplasmic contaminations and particularly, of cytoplasmic 80S ribosomes. VEE RNA when added to submitochondrial system caused significant stimulation of RNA and protein synthesis. These processes were resistant to actinomycin D which inhibited profoundly the synthesis of proper mitochondrial macromolecules. The stimulating effect of VEE RNA in experiments with submitochondrial system was about three times higher than that with intact mitochondria. The stimulation of¹⁴C-amino acid incorporation increased as a function of incubation time; a certain lag-period being observed. The newly formed virus-specific RNA's and ribonucleoproteins were identified with the aid of sedimentation analysis. In particular, radioactive RNA's with sedimentation coefficients 40S and 26-18S were isolated from the incubated system. These RNA's are similar respectively to VEE genome RNA and doublestranded VEE replicative RNA. In double labelling experiments with³H-uridine and¹⁴Camino acids it was shown that VEE RNA induced synthesis of ribonucleoproteins containing newly formed RNA and protein. These RNP possessed sedimentation coefficients 60-80S, 140S and 300S in sucrose gradient and buoyant densities 1.32 and 1.50 g/cm³ in cesium chloride gradients. These properties of ribonucleoproteins synthesized de novo in submitochondrial system are close to those of RNP intermediates of VEE virus reproduction in the infected cells. We concluded that viral RNA could program virus-specific synthesis in the submitochondrial system under conditions that eliminated the contribution of cytoplasmic ribosomes.     

On the distribution of the nucleotides in seven completely sequenced DNAs

Several Markov chain models (up to fourth order) have been fitted to the sequences of the seven DNAs presented in Fuchs et al. (1980). Two methods for determining the order of Markov chain are applied to the data. The two methods lead to different conclusions and we dicuss these discrepancies. When the distribution of the nucleotides in a DNA sequence is investigated, it is suggested that the study on the order of the Markov model should be supplemented with additional analysis.     

On the distribution of the nucleotides in the seven completely sequenced DNAs

Several Markov chain models (up to fourth order) have been fitted to the sequences of the seven DNAs presented in Fuchs et al. (1980). Two methods for determining the order of Markov chain are applied to the data. The two methods lead to different conclusions and we dicuss these discrepancies. When the distribution of the nucleotides in a DNA sequence is investigated, it is suggested that the study on the order of the Markov model should be supplemented with additional analysis.     

The role of nonreal eigenvalues in the identification of cycles in a compartmental system

The paper concerns the relationship between the cycles in the graph of a compartmental system and the modes of the impulse response function associated with an input-output experiment. Suppose that there is at least one oscillatory mode, e^μtcos(vt - α). Let e^?t be the slowest mode. The main result is that the system contains a cycle of length 3 or longer and that the length of the longest cycle is at least π/tan^-1[¦v¦/(?-μ)]. The paper also deals with the problem of estimating the cycle length from discrete data.     

Complexity of compartmental models

A family of complexity indices for compartmental models is introduced. These are analogous to the diversity indices used to study ecological communities. Each index is shown to be related to the eigenvalues of the model, and in one case (the Shannon index) is equal to the negative of the sum of all eigenvalues. Relations between these indices and equivalent models as well as intrinsic complexity are investigated. In each case, large values of the complexity index lead to greater stability, but not necessarily conversely.     

Some equivalent compartmental models

Compartmental models with strongly connected digraphs always have a stable equilibrium solution. Each such model may be reduced by a sequence of matrix operations to a mammilary system with the same equilibrium solution. It may also be reduced by a sequence of operations on the digraph. It may then be further reduced to a model whose digraph is a 2-cycle with the same mean first passage time m. If this m is taken as an indicator of the relative stability, then the latter is independent of the complexity as measured by the number of arcs per vertex. In a number of special cases the index m is related to the eigenvalues of the matrix. It is shown to have a simple relation to other time parameters as well.     

Structural properties of compartmental models

This article treats structural properties of the inverse of a compartmental matrix and how they relate to properties of coefficients of the transfer function of the compartmental system. Newly formulated conditions are presented for certain of these parameters to be zero or positive. Also results are given on the interdependence of transfer function coefficients and how this relates to the identifiability problem. Answers to some questions raised in the recent literature about coefficient dependence are discussed.