Analysis of tracer experiments: VI. Determination of partitioned initial entry functions

It is shown that the partitioned initial entry functions previously introduced in multicompartment analysis can be directly and uniquely determined from the experimental data.     

Note on oscillations in a closed chain of compartments with linear,unidirectional transport

A closed chain of compartments in which there is unidirectional transport between adjacent members can exhibit damped oscillations. For a system ofn equivalent compartments, the value ofn which gives the greatest difference between the first maximum and first minimum isn=11, the difference being 1.57%. The greatest difference between the first maximum value and the steady state value is 4% and is obtained whenn=25. The results are illustrated graphically forn equal to 5, 10, 25 and 100.     

An integral equation approach to measuring turnover in nonsteady compartmental and distributed systems

Physiological systems are often modelled by a set of compartments. Alternatively they can be described by the diffusion-convection-reaction equations governing distributed systems. The problem considered here is that of identifying a continuously changing input of some metabolite )tracee), endogenous to the system and hence inaccessible, when a nonlinear or time-varying component is also introduced into the loss parameter, as for example through feedback mechanisms. A tracer is used to determine the steady-state impulse response under time-invariant, linear conditions. A known input of tracer is also administered when the system is driven out of steady state. The integral equations developed utilize the predetermined impulse response, the measured concentrations of both tracer and tracee (output) in some region of the system to estimate the changing loss parameter and the unknown input in a continuous fashion.     

Generation of nonidentical compartments in vesicular transport systems

How can organelles communicate by bidirectional vesicle transport and yet maintain different protein compositions? We show by mathematical modeling that a minimal system, in which the basic variables are cytosolic coats for vesicle budding and membrane-bound soluble N-ethyl-maleimide–sensitive factor attachment protein receptors (SNAREs) for vesicle fusion, is sufficient to generate stable, nonidentical compartments. A requirement for establishing and maintaining distinct compartments is that each coat preferentially packages certain SNAREs during vesicle budding. Vesicles fuse preferentially with the compartment that contains the highest concentration of cognate SNAREs, thus further increasing these SNAREs. The stable steady state is the result of a balance between this autocatalytic SNARE accumulation in a compartment and the distribution of SNAREs between compartments by vesicle budding. The resulting nonhomogeneous SNARE distribution generates coat-specific vesicle fluxes that determine the size of compartments. With nonidentical compartments established in this way, the localization and cellular transport of cargo proteins can be explained simply by their affinity for coats.     

Qualitative theory of compartmental systems with lags

Dynamic models of many processes in the biological and physical sciences give systems of ordinary differential equations called compartmental systems. Often, these systems include time lags; in this context, continuous probability density functions (pdfs) of lags are far more important than discrete lags. There is a relatively complete theory of compartmental systems without lags, both linear and non-linear [SIAM Rev. 35 (1993) 43]. The authors extend their previous work on compartmental systems without lags to show that, for discrete lags and for a very large class of pdfs of continuous lags, compartmental systems with lags are equivalent to larger compartmental systems without lags. Consequently, the properties of compartmental systems with lags are the same as those of compartmental systems without lags. For a very large class of compartmental systems with time lags, one can show that the time lags themselves can be generated by compartmental systems without lags. Thus, such systems can be partitioned into a main system, which is the original system without the lags, plus compartmental subsystems without lags that generate the lags. The latter may be linear or non-linear and may be inserted into main systems that are linear or non-linear. The state variables of the compartmental lag subsystems are hidden variables in the formulation with explicit lags.     

Localized RNAs and their functions

The eukaryotic cell is partitioned by membranes into spatially and functionally discrete subcellular organelles. In addition, the cytoplasm itself is partitioned into discrete subregions that carry out specific functions. Such compartmentation can be achieved by localizing proteins and RNAs to different subcellular regions. This review will focus on localized RNAs, with a particular emphasis on RNA localization mechanisms and on the possible biological functions of localization of these RNAs. In recent years, an increasing number of localized RNAs have been identified in a variety of cell types among many animal species. Emphasis here will be on localized RNAs in the most intensively studied systems – Drosophila and Xenopus eggs and early embryos.     

Proliferation-differentiation pathways of murine haemopoiesis: correlation of lineage fluxes

Abstract. A quantitative analysis of literature data on muiine bone marrow haemopoiesis was performed in order to determine how the haemolpoietic system fulfils requirements of cell turnover in the normal steady state. In particular, the production rates (fluxes) of erythrocytes, megakaryocytes, monocytes, neutrophils, eosinophils, basophils and mast cells have been calculated and normalized. Subsequently, based on available data on relationships of various lineages, a working model of the developmental schema of haemopoiesis was derived. Since, according to the principle of conservation, the effluxes of all lineages from bone marrow in the steady state must be equal to effluxes from the compartments of respective precursors divided by the coefficient of multiplication, the correlation of lineage fluxes was used to determine the position of branching points. It was concluded that all the quantita.tive data on lineage proliferation in the normal steady state in vivo, available for analysis, are fully consistent with a simple binary model of the sequential type, similar to that suggested by Brown and coworkers (Brown G., Jones NA, Bunce CM, Owen PJ, IPatton WN. (1988) Haemopoiesis: a lottery or determinism?Differentiation, 39 , 83).     

An optimizing start-up strategy for a bio-methanator

This paper presents an optimizing start-up strategy for a bio-methanator. The goal of the control strategy is to maximize the outflow rate of methane in anaerobic digestion processes, which can be described by a two-population model. The methodology relies on a thorough analysis of the system dynamics and involves the solution of two optimization problems: steady-state optimization for determining the optimal operating point and transient optimization. The latter is a classical optimal control problem, which can be solved using the maximum principle of Pontryagin. The proposed control law is of the bang–bang type. The process is driven from an initial state to a small neighborhood of the optimal steady state by switching the manipulated variable (dilution rate) from the minimum to the maximum value at a certain time instant. Then the dilution rate is set to the optimal value and the system settles down in the optimal steady state. This control law ensures the convergence of the system to the optimal steady state and substantially increases its stability region. The region of attraction of the steady state corresponding to maximum production of methane is considerably enlarged. In some cases, which are related to the possibility of selecting the minimum dilution rate below a certain level, the stability region of the optimal steady state equals the interior of the state space. Aside its efficiency, which is evaluated not only in terms of biogas production but also from the perspective of treatment of the organic load, the strategy is also characterized by simplicity, being thus appropriate for implementation in real-life systems. Another important advantage is its generality: this technique may be applied to any anaerobic digestion process, for which the acidogenesis and methanogenesis are, respectively, characterized by Monod and Haldane kinetics.     

Equilibrium and steady state thermodynamics of active transport systems studied on simple models simulating Ca2+ transport through sarcoplasmic reticulum membranes

Equilibrium and steady state conditions of primary active transport systems are analyzed in models simulating well known characteristics of calcium transport through sarcoplasmic reticulum membranes. The model for the equilibrium simulations is a closed system with two compartments and a vectorial chemical reaction coupling Ca transport and ATP breakdown. The chemical potential difference for Ca (delta mu Ca) is calculated as a function of the total amount of Ca (Cat) and nucleotides (Nt) in the system. Results are obtained by successive approximations along the thermodynamic pathway of the reaction, up to minimizing free energy of the system, since the solution of the explicit equations cannot be obtained with computers of current precision for data within physiological ranges. delta mu Ca and [Caout] are extremely dependent on Cat and Nt for certain combinations of the variables, i.e. [Caout] can be raised from 10(-8) to 10(-6) M when Cat varies from 0.998 to 1.002 mM, therefore, the running force of the spontaneous reaction is largely shifted by tiny changes in the parameters of the system. For steady state simulations, ATP supply to the system, ADP and Pi drainage, and Ca diffusion through the barrier, are assumed. Again, conditions within physiological ranges can be found where tiny changes in Cat, the rate of ATP supply, diffusion, the ratio between the volumes of the compartments, or a relative uncoupling between the transport and hydrolytic reactions, largely shifts delta mu Ca and [Caout], thus making the steady state highly unstable and therefore well designed to operate as an amplifier of physiological signals. The equilibrium model describes some physicochemical characteristics of the system; the steady state model is more useful to simulate several physiological situations.     

Optimal dynamic experiments for bioreactor model discrimination

 This paper describes a general approach fordynamic model discrimination for continuous cultures and presents dynamic models for pure cultures of E. coli and C. utilis obtained using the method. For each pure culture system, four candidate models representing various levels of structure were considered. All models reduce to Monod growth kinetics at steady state. An optimized set of multivariable step inputs in selected manipulative variables was used to discriminate between candidate models. The models that best predicted the dynamic behavior were selected by comparison of model predictions with experimental data. Two discrimination functions were compared in terms of their ability to determine the optimal set of multivariable step inputs to discriminate between candidate models. Results indicate that model discrimination based on maximizing the minimum absolute difference between any two models for a given set of inputs possessed good potential for discrimination between candidate models. Models selected for E. coli andC. utilis from the model discrimination work arepresented and compared with experimental data. Received: 24 May 1994/Received revision: 28 September 1994/Accepted: 5 December 1994     

Linear systems approach to analysis of complex dynamic behaviours in biochemical networks

Central functions in the cell are often linked to complex dynamic behaviours, such as sustained oscillations and multistability, in a biochemical reaction network. Determination of the specific mechanisms underlying such behaviours is important, e.g. to determine sensitivity, robustness, and modelling requirements of given cell functions. In this work we adopt a systems approach to the analysis of complex behaviours in intracellular reaction networks, described by ordinary differential equations with known kinetic parameters. We propose to decompose the overall system into a number of low complexity subsystems, and consider the importance of interactions between these in generating specific behaviours. Rather than analysing the network in a state corresponding to the complex non-linear behaviour, we move the system to the underlying unstable steady state, and focus on the mechanisms causing destabilisation of this steady state. This is motivated by the fact that all complex behaviours in unforced systems can be traced to destabilisation (bifurcation) of some steady state, and hence enables us to use tools from linear system theory to qualitatively analyse the sources of given network behaviours. One important objective of the present study is to see how far one can come with a relatively simple approach to the analysis of highly complex biochemical networks. The proposed method is demonstrated by application to a model of mitotic control in Xenopus frog eggs, and to a model of circadian oscillations in Drosophila. In both examples we are able to identify the subsystems, and the related interactions, which are instrumental in generating the observed complex non-linear behaviours.     

Nonsteady-State Three Compartment Tracer Kinetics: I. Theory

A set of differential equations is derived which describes the four unidirectional fluxes of a substance across the boundaries of the central compartment of a serially arranged three compartment system, and the amount of this substance present in the central compartment. An analytic solution is obtained which yields all of these quantities as functions of time. The analysis is associated with a defined set of repetitive experiments from which the necessary data are obtained and during which the two outer compartments must be subject to experimental control. The solution is applicable to both the initial steady state and a transient, time-dependent state created by making a step change in the initial conditions. It describes the fluxes and compartment size without assuming that constant kinetic coefficients relate the fluxes to compartmental quantities but is limited by the requirement that the response of the system be repeatable in time.     

A stochastic compartmental model with continuous infusion

This paper deals with stochasticm-compartmental systems with continuous time-dependent infusions into all compartments and reversible time-independent flows between any two compartments. A methodology for the first two moments of the distribution of the number of units in the different compartments at any point in time is outlined without resorting to the usual techniques of generating functions and inverse Laplace transforms. A possible application to a systems analysis of the kidney transplant system is discussed.     

Logical identification of all steady states: The concept of feedback loop characteristic states

Biological regulatory systems can be described in terms of non-linear differential equations or in logical terms (using an “infinitely non-linear” approximation). Until recently, only part of the steady states of a system could be identified on logical grounds. The reason was that steady states frequently have one or more variable located on a threshold (see below); those steady states were not detected because so far no logical status was assigned to threshold values. This is why we introduced logical scales with values 0,¹θ, 1²θ, 2, ..., in which¹θ,²θ, ... are the logical values assigned to the successive thresholds of the scale. We thus have, in addition to the regular logical states,singular states in which one or more variables is located on a threshold. This permits identifyingall the steady states on logical grounds. It was noticed that each feedback loop (or reunion of disjointed loops) can be characterized by a logical state located at the thresholds at which the variables of the loop operate. This led to the concept ofloop-characteristic state, which, as we will see, enormously simplifies the analysis.The core of this paper is a formal demonstration that among the singular states of a system, only loop-characteristic states can be steady. Reciprocally, given a loop-characteristic state, there are parameter values for which this state is steady; in this case, the loop is effective (i.e. it generates multistationarity if it is a positive loop, homeostasis if it is a negative loop). This not only results in the above-mentioned radical simplification of the identification of the steady states, but in an entirely new view of the relation between feedback loops and steady states.     

Modelling the production and transport of dissolved organic carbon in forest soils

DyDOC describes soil carbon dynamics, with a focus on dissolved organic carbon (DOC). The model treats the soil as a three-horizon profile, and simulates metabolic carbon transformations, sorption reactions and water transport. Humic substances are partitioned into three fractions, one of which is immobile, while the other two (hydrophilic and hydrophobic) can pass into solution as DOC. DyDOC requires site-specific soil characteristics, and is driven by inputs of litter and water, and air and soil temperatures. The model operates on hourly and daily time steps, and can simulate carbon cycling over both long (hundreds-to-thousands of years) and short (daily) time scales. An important feature of DyDOC is the tracking of ¹⁴C, from its entry in litter to its loss as DO¹⁴C in drainage water, enabling information about C dynamics to be obtained from both long-term radioactive decay, and the characteristic ¹⁴C pulse caused by thermonuclear weapon testing during the 1960s ("bomb carbon"). Parameterisation is performed by assuming a current steady state. Values of a range of variables, including C pools, annual DOC fluxes, and ¹⁴C signals, are combined into objective functions for least-squares minimisation. DyDOC has been applied successfully to spruce forest sites at Birkenes (Norway) and Waldstein (Germany), and most of the parameters have similar values at the two sites. The results indicate that the supply of DOC from the surface soil horizon to percolating water depends upon the continual metabolic production of easily leached humic material. In contrast, concentrations and fluxes of DOC in the deeper soil horizons are controlled by sorption processes, involving comparatively large pools of leachable organic matter. Times to reach steady state are calculated to be several hundred years in the organic layer, and hundreds-to-thousands of years in the deeper mineral layers. It is estimated that DOC supplies 89% of the mineral soil carbon at Birkenes, and 73% at Waldstein. The model, parameterised with "steady state" data, simulates short-term variations in DOC concentrations and fluxes, and in DO¹⁴C, which are in approximate agreement with observations.     

An approximate analytical solution to the lag period of monophenolase activity of tyrosinase

Tyrosinase shows a lag period in its action on monophenols (l-tyrosine). We propose an approximate analytical solution for the lag period, which fulfils the dependences with regard to initial enzyme concentration, and initial monophenol concentration. Furthermore, from a study of the dependences of the lag period on these variables, we can determine experimentally the o-diphenol concentration in the steady state. The Michaelis constant of the o-diphenol in the presence of the monophenol can be determined from the relationship between the o-diphenol concentration in the steady state and the initial monophenol concentration, taking into consideration the experimentally calculated Michaelis constant for the monophenol substrate. Although this Michaelis constant is much lower than the Michaelis constant for diphenol in the absence of monophenol, the binding site is the same. A kinetic analysis of the action mechanism of tyrosinase explains this difference in the values of the Michaelis constants.     

Trafficking motifs as the basis for two-compartment signaling systems to form multiple stable states

Transport of molecules in cells is a central part of cell biology. Frequently such trafficking is not just for material transport, but also for information propagation, and serves to couple signaling circuits across cellular compartments. Here, I show that trafficking transforms simple local signaling pathways into self-organizing systems that span compartments and confer distinct states and identities to these compartments. I find that three motifs encapsulate the responses of most single-compartment signaling pathways in the context of trafficking. These motifs combine with different trafficking reactions to generate a diverse set of cellular functions. For example, trafficked bistable switches can oscillate or become quad- or tristable, depending on trafficking mechanisms and rates. Furthermore, the analysis shows how compartments participating in traffic can settle to distinct molecular compositions characteristic of distinct organelle identities. This general framework shows how the interplay between molecular movement and local reactions can generate many system functions, and give distinct identities to different parts of the cell.     

Analyzing steady states of dynamics of bio-molecules from the structure of regulatory networks

Regulatory relations between biological molecules constitute complex network systems and realize diverse biological functions through the dynamics of molecular activities. However, we currently have very little understanding of the relationship between the structure of a regulatory network and its dynamical properties. In this paper we introduce a new method, named “linkage logic” to analyze the dynamics of network systems. By this method, we can restrict possible steady states of a given complex network system from the knowledge of regulatory linkages alone. The regulatory linkage simply specifies the list of variables that affect the dynamics of each variable. We formalize two aspects of the linkage logic: the “Principle of Compatibility” determines the upper limit of the diversity of possible steady states of the dynamics realized by a given network; the “Principle of Dependency” determines the possible combinations of states of the system. By combining these two aspects, (i) for a given network, we can identify a cluster of nodes that gives an alternative representation of the steady states of the whole system, (ii) we can reduce a given complex network into a simpler one without loss of the ability to generate the diversity of steady states, (iii) we can examine the consistency between the structure of network and observed set of steady states, and (iv) sometimes we can predict unknown states or unknown regulations from an observed set of steady states alone. We illustrate the method by several applications to an experimentally determined regulatory network for biological functions.     

An insight into tumor dormancy equilibrium via the analysis of its domain of attraction

The trajectories of the dynamic system which regulates the competition between the populations of malignant cells and immune cells may tend to an asymptotically stable equilibrium in which the sizes of these populations do not vary, which is called tumor dormancy. Especially for lower steady-state sizes of the population of malignant cells, this equilibrium represents a desirable clinical condition since the tumor growth is blocked. In this context, it is of mandatory importance to analyze the robustness of this clinical favorable state of health in the face of perturbations. To this end, the paper presents an optimization technique to determine whether an assigned rectangular region, which surrounds an asymptotically stable equilibrium point of a quadratic systems, is included into the domain of attraction of the equilibrium itself. The biological relevance of the application of this technique to the analysis of tumor growth dynamics is shown on the basis of a recent quadratic model of the tumor–immune system competition dynamics. Indeed the application of the proposed methodology allows to ensure that a given safety region, determined on the basis of clinical considerations, belongs to the domain of attraction of the tumor blocked equilibrium; therefore for the set of perturbed initial conditions which belong to such region, the convergence to the healthy steady state is guaranteed. The proposed methodology can also provide an optimal strategy for cancer treatment.