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.     

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.     

Eigenvalues and structure of compartmental models

The “spread” of the nonzero eigenvalues of a compartmental matrix is studied by reference to the associated directed graph. It is related to the eigenvalues of the matrices of the individual cycles for certain strongly connected directed graphs. The equilibrium solution to the entire model is also an equilibrium solution to the model consisting of the individual cycles.     

Analysis of compartmental models of ligand-induced endocytosis

Kinetic models have played a pivotal role in the study of ligand-induced endocytosis. However, an analysis that suggests a systematic way to validate such models is lacking. The current work analyses the base model of ligand-induced endocytosis for three widely used experimental protocols. In protocol I cells initially devoid of ligand are incubated in ligand solution, whereas protocols II and III are desorption experiments in which an initial pool of surface or internalized ligand-receptor complexes, respectively, are released into an elution medium that is initially devoid of ligand. A short-time analysis of protocol I using successive substitutions yielded a corrected pre-factor for the In/Sur plot introduced by Wiley and Cunningham (Cell 25 (1981) 433). In contrast, neglecting the variation in receptor numbers yielded an approximation of protocol I that is valid for long times (e.g. tens of minutes). Similarly, the low cell-concentration limits of protocols II and III are derived by neglecting the concentration of free ligand. The simplicity of these approximations provides a simple and reliable method for estimating the parameters governing ligand kinetics, while their definitive nature implies that they can be used to verify the validity of the base model. This analysis also provides insight on the fast endocytosis and recycling limit of protocol III.     

Parameter space boundaries for unidentifiable compartmental models

Methods for dealing with unidentifiable compartmental models are first reviewed, emphasizing the parameter interval analysis and exhaustive modeling approaches. More general methods are presented for generating the set of all nonnegative parameter solutions that localize the parameters within bounded regions of parameter space and extend previously published parameter bounding strategies. Each point of these regions is an equivalent solution of the parameter identification problem. If a point on the boundary is selected, at least one of the parameters vanishes and an equivalent submodel is obtained. This property shows the close relationship between the exhaustive modeling and parameter interval analysis approaches.     

Properties of the transfer functions of compartmental models

This paper is concerned with the nonlinear system of algebraic equations relating the positive parameters of a linear time-invariant compartmental model to its transfer function coefficients. The general form that these equations must take is shown, and simple necessary conditions for the existence of positive solutions are given. An immediate use of these conditions is the development of necessary conditions for a polynomial with positive coefficients to have negative roots. A method is then outlined which triangularizes the system and reduces the complete solution problem to one of finding and counting roots of a polynomial. Sufficient conditions for the existence of real and positive solutions are demonstrated.     

Equilibrium points for nonlinear compartmental models.

Equilibrium points for nonlinear autonomous compartmental models with constant input are discussed. Upper and lower bounds for the steady states are derived. Theorems guaranteeing existence and uniqueness of equilibrium points for a large collection of system are proved. New information relating to mean residence times is developed. Asymptotic results and a section on stability are included. A recursive process is discussed that generates iterates that converge to steady states for certain types of models. An interesting range of models are included as examples. An attempt is made to provide general qualitative theory for such nonlinear compartmental systems.     

The relation between Krogh and compartmental transport models

The meaning of compartmental concentration is intuitively simple in well-stirred compartment models with uniform concentration but not so in the general case. This meaning may be better understood by relating the general compartmental model to a spatially explicit one such as the Krogh cylinder model. We show that compartmental equations result directly by spatial averaging of the Krogh partial differential equations. Intercompartmental transport is usually stated as flux is proportional to the capillary-tissue concentration difference or $\text{J = ? (}\text{C}{}_{\mathrm{c}}\text{-}\text{C}{}_{\mathrm{T}}\text{)}$. We verify this simple equation by showing that flux in the Krogh model is also approximately first order and then derive the transport coefficient, ?, in terms of the geometry and diffusivity in the Krogh model. A relation between capillary and venous concentration or, for a gas, partial pressure is needed. For the highly diffusable, metabolic gas C0₂ this relation is $\text{P}{}_{\mathrm{c}}\text{= P}{}_{\mathrm{v}}\text{?}\text{M}\text{20K}{}_{\mathrm{c}}$ that is, capillary and venous partial pressures differ by one half the metabolic generation.     

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.     

Stochastic compartmental models with banching and immigrant particles

A multicompartmental model in which particles enter the system from the environment and reproduce according to a Markov branching process has been considered. Explicit expressions have been obtained for the mean vector and the correlation structure for the numbers of particles in different compartments in different time points of the system. Growth rates of the mean vector and some special cases of the system are also discussed.     

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.     

Some compartmental models of the root: steady-state behavior

Plots of the pressure difference (DeltaP) applied to plant roots vs. the resulting volume flow rate (Q(v)) often exhibit an anomalous offset that has been difficult to explain. The present analysis suggests that solute build-up in two- and three-compartment models of the root cannot account for this offset. The Ginsburg-Newman three-compartment model explains the offset in terms of differing reflection coefficients for the membranes bounding the intermediate compartment. This model appears more promising, but it predicts a minimum in the plot of xylem-sap osmotic pressure vs. Q(v)which is not observed in practice. Fiscus hypothesized that an internal asymmetric distribution of non-mobile solutes is responsible for the offset. In the present study, this hypothesis is incorporated into a four-compartment model of the root that is conceptually related to the three-compartment model of Miller. But according to the four-compartment model, the asymmetric solute distribution does not arise because of solvent drag. Rather the anomalous offset is associated with a concentration gradient of photoassimilates (the non-mobile solutes) that exists in the absence of volume flow, and which drives the diffusive transport of these solutes from the stele to the cortex via endodermal plasmodesmata. This model is consistent with the existence of radial symplastic osmotic-pressure gradients, and it appears to have greater explanatory power than the Ginsburg-Newman model. In particular, it suggests explanations for diurnal variations in DeltaP-Q(v)curves, as well as the effects of changing external solute concentrations. It also shows how the overall root reflection coefficient can be less than unity, even when the cell membranes are effectively ideally semipermeable, and there is negligible extracellular transport of water and solutes. The model makes a number of experimentally testable predictions.     

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.     

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.     

Using extracellular action potential recordings to constrain compartmental models

We investigate the use of extracellular action potential (EAP) recordings for biophysically faithful compartmental models. We ask whether constraining a model to fit the EAP is superior to matching the intracellular action potential (IAP). In agreement with previous studies, we find that the IAP method under-constrains the parameters. As a result, significantly different sets of parameters can have virtually identical IAP’s. In contrast, the EAP method results in a much tighter constraint. We find that the distinguishing characteristics of the waveform—but not its amplitude- resulting from the distribution of active conductances are fairly invariant to changes of electrode position and detailed cellular morphology. Based on these results, we conclude that EAP recordings are an excellent source of data for the purpose of constraining compartmental models. Action Editor: Alain Destexhe     

Passage time,resilience, and structure of compartmental models

The relation between the graphical structure and two measures of stability of a compartmental model is studied. The two are resilience as measured by deviation from equilibrium throughout the history of the system and mean first passage time from a central compartment back to itself. A number of examples are studied and a theory developed for models whose graphical structure is in the form of an advanced rosette. The resilience, known to increase with complexity, is shown to increase as well with uneveness in length of food chains. The mean first passage time, on the other hand, depends only on the number of food chains and their average length.     

Metabolism of human apolipoproteins A-I and A-II: compartmental models

The metabolism of radioiodinated apolipoproteins (apo) A-I and A-II have been examined using the techniques of compartmental modeling. The model for apoA-I contains two plasma compartments decaying at different rates. One component of apoA-I has a residence time of 3.8 days and the second has a residence time of 6.1 days. In contrast, the apoA-II model has only one plasma component, with a residence time of 5.5 days, which decays through two distinct pathways. Twenty-seven percent of apoA-II decays through a pathway that takes 1.1 days longer to reach the urine than the remaining 73% which decays through the more direct path. These differences in the metabolism exist in both male and female populations. Comparison of fasting and nonfasting concentrations of apoA-I revealed that apoA-I concentration was elevated 0.5 standard deviations in the nonfasting samples while there was no significant difference in the apoA-II concentrations. The fasting apoA-I concentrations were found to be less stable over the study period when compared to fasting apoA-II concentrations. These findings are interpreted as indicating that apoA-I and apoA-II each have a separate metabolism which overlaps when they are present on the same lipoprotein particle. Furthermore, these findings are consistent with the concept that apoA-I metabolism is influenced more by perturbations such as dietary modulation.