A network model of the coupling of ion channels with secondary messenger in cell signalling

We demonstrate that formal neural networks techniques allow to build the simplest models compatible with a limited but systematic set of experimental data. The experimental system under study is the growth of mouse macrophage like cell lines under the combined influence of two ion channels, the growth factor receptor and adenylate cyclase. We conclude that 3 components out of 4 can be described by linear multithreshold automata. The remaining component behavior being non-monotonous necessitate the introduction of a fifth hidden variable, or of non-linear interactions. P.A.C.S.: 87.10 General, theoretical and mathematical biophysics 87.25 Cellular biophysics Short title: A network model of the coupling of ion channels Preprint L.P.S. 363, February 1992 Appeared in Network: Computation in Neural Systems Volume 3, Number 4 (November 1992), pp. 393-406. Introduction A single cell viewed as a set of reacting chemical species is already a rather complex system. In fact the set of interacting genes and products in the cell was the object of one of the first modelisation of a living system in terms of boolean automata by S. Kauffman (1969) and by R. Thomas (1975). At the cell membrane level, biologically active compounds (growth factors, hormones, neurotransmitters, immunoreactive compounds, etc.) interact with specific receptor proteins. These "first messengers" interact with the receptor and induce a sequence of biochemical events which works like a channel of information transfer. At the end of the process, the cell performs the biological effect triggered by the signal: DNA synthesis and cell division by growth factors, nerve or muscle activity by neurotransmitters, turning on synthesis of given enzymes or activating given metabolic reactions by hormones. Following the interaction of first messengers with their receptors, systems of "second messengers" transfer the signal from the membrane to target molecular structures in the cytoplasm of the nucleus. However, as discussed recently, rather than linear sequences of individual molecular events, the mechanisms involved must be analyzed as dynamic changes in the cellular state as in state machines (Lichstein & Atlan 1990). The number of second messengers is very small as compared to the number of structurally different receptors, each responsible specifically for a given biological effect. Therefore, in order to retain specificity at the level of the cytoplasmic or nuclear target, the second messenger must react in a complex manner with internalized or activated receptor associated membrane proteins which would serve as modulators of its activity. An example of a non specific activation of a receptor associated protein participating in a specific message may be found in observations on cell division induced by growth factors (Paris & Pouyssegur, 1986; Panet et al., 1986a,b; Pouyssegur et al., 1982; Berridge, 1986; Panet & Atlan, 1990, 1991). It was shown that growth factors induce an increase in both intracellular Ca, pH and possibly Na, and all of these seem necessary to induce de novo DNA synthesis. Two ion transporters associated with the growth factor receptors (Na-H antiport and Na, K, Cl cotransport) seem to participate, at least as modulators, in the transfer of information by the second messenger. Thus, second messengers with their modulators, although not very specific by themselves, can be used in combinations by the cell to confer the specificity of the response. Evidence was presented (Rapp & Berridge, 1977) on coupling between cytoplasmic calcium and cAMP through common regulatory loops. Rapp & Berridge (1977) proposed mathematical models of instabilities leading to oscillations of the Ca and cAMP concentrations responsible for oscillatory behavior of several cellular properties. However, in general, oscillations are not the only possible outcome of such coupling. Sets of different steady-states can also be established. More recently, a spatial-temporal model of cell activation has been proposed (Alkon & Rasmussen, 1988) taking into account multiple coupling (both in series and in parallel) between different components of Ca + dependent second messenger, cAMP, ion channels and receptors. In most of these observations, the complexity of the interactions in the form of various couplings between different reactions, even for only one well-defined cellular function, often prevents a direct interpretation of available experimental results. System dynamics methods of analysis are necessary to help build up schemes of such interactions that would account for the observed cellular states. In fact, the state of a cell, as that of state machine (see Holcombe, 1982:66-71 for state machine representations of metabolic pathways), is the set of concentrations of its constitutive molecular species at every point in time and space co-ordinates. The change from one state to the other is governed by the chemical and transport kinetics of tens or hundreds of coupled reactions. Studies of such kinetics of relatively simple chemico-diffusional networks, involving not more than a few coupled reactions (Holcombe, 1982; Oster et al., 1973; Mickulecky, 1977; Atlan et al., 1979), have shown already that a change in the concentration of a compound somewhere in the metabolic network is likely to affect simultaneously the concentration of many other compounds elsewhere. Highly non-linear effects (excitatory or inhibitory) can be produced by feedback inhibition or autocatalytic loops, so that the classical idea of being able to modify one parameter at a time is generally wrong. The usual methods of quantitative analysis are those of reaction kinetics making use of sets of differential or partial differential equations. Network Thermodynamics (Oster & al., 1973) has been presented as a discrete representation, a priori more convenient for physicochemical couplings involving different energy domains. However, when confronted with actual experimental data on such coupled reactions obtained under in vivo conditions, e.g. of cell cultures, these methods are not of a great help because they need much more detailed data (binding constants, reaction rates, concentration changes, etc.) than what is available. The purpose of this paper is to show that the techniques developed in learning by neural nets can be used to analyze sets of complex data from experiments on in vivo cellular systems. They will allow us to determine a set of interactions among receptors, ion transporters and second messengers that interact in the built up of a proliferative response to serum growth factors. The method is further called PAWN from the initials of the authors. 1. The experimental system In a series of experiments, R. Panet, H. Atlan and co-workers studied the influence of ions transporters on the proliferation of different cell lines (mouse fibroblasts) and of human fibroblasts in culture (Panet & Atlan, 1990, 1991). In parallel, making use of cell lines of macrophages with mutants deficient in adenylate cyclase, A. Bourrit, H. Atlan et al. were able to show the influence of growth factor receptors, of adenylate cyclase, the enzyme implied in the synthesis of cAMP (a secondary messenger) and of two different transporters of Na and K ions, one ouabain sensitive (OS) and the other one bumetanide sensitive (BS) (this means that they can be blocked respectively by ouabain or bumetanide). In other words, the two ion transporters and the enzyme play a role in the transduction of the primary message, the linking of the growth factor, into cell proliferation. But no simple linear scheme of interactions explains all the experimental results. Binary interactions seem to be sometimes positive, sometimes negative, and retroaction loops are likely. In order to elucidate the question, they did 12 independent experiments on the same system, i.e. the above-mentioned lines of macrophages. In each experiment but one, some of the four components of the system (growth factor receptor, adenylate cyclase, OS Na K pump and BS NaKCl cotransporter were absent or inhibited (and thus clamped at null activity), and the activities of the nonclamped components were measured. . One observes, for each component, from 2 to 4 (depending on the component) well distinct, reproducible, levels of activity. In the following, disregarding the precise values of the activities, we will only take into account the existence of different (and ordered) activity levels (Table 1), and propose the simplest possible model which is consistent with the experimental results. Component: I II III IV Exp. n° 01. ...........0 1 2 3 02. ...........2 1 0 1 03. ...........3 0 1 2 04. ...........3 0 0 1 05. ...........2 0 1 0 06. ...........2 0 0 0 07. ...........1 1 2 0 08. ...........1 1 0 0 09. ...........0 0 1 1 10. ...........0 0 0 1 11. ...........0 1 2 3 12. ........... 0 1 0 0 Table 1. Discretized activities of each component in the 12 experiments. Underlined zeros correspond to clamped activities. Let us first comment the data of Table 1. Element I represents the bumetanide sensitive transporter. It can be clamped to zero by the presence of bumetanide in the culture, which block the activity of this Na/ K/ Cl co-transporter. The (measured) activities can take 4 possible discretised levels that we note 0, 1, 2 or 3 (corresponding resp. to experimental ranges: 0-2; 7; 12-13; 15-17; in mol./10 min. for 10 6 cells). Element II represents the growth factor receptor activity which is set to 0 in the absence of the growth factor (serum deprived cells) and 1 in its presence. Element III is the adenine cyclase activity. This activity can have 3 possible levels, 0 , 1 or 2, (corresponding resp. to mutants deficient in the enzyme, to normal level in non mutants in the absence of growth factor or to activity stimulated by serum growth factors). Element IV is the oubain sensitive Na/K transporter, which can be clamped to 0 by the presence of oubain. The measured activities can take 4 possible levels 0, 1, 2 or 3 (corresponding resp. to experimental ranges: 0-2; 18-26; 33-35; 39-43; in mol./10 min. for 10 6 cells). The total number of independent experiments is in principle 16=2 4 , since each element can be either set free or clamped, but clamping all elements would bring no new information. Furthermore, since the authors were interested in the influence of growth factors on ionic transporters, the experiments where both would be clamped to zero were not done. This explains why the total number of experiments is limited to 12. 2. The PAWN method: theoretical framework A simple theoretical framework compatible with the scarcity and precision of experimental results is that of network of threshold automata. Binary threshold automata are better known as formal neurons (Hertz et al. 1990). We here use a more general formalism, multithreshold automata (Thomas 1990). The "field" h i acting on automaton i is defined by: