# Cellular systems—II. stability of compartmental systems

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<ul><li><p>Cellular Systems-II. Stability of Compartmental Systems* </p><p>G. S. LADDE </p><p>Department of Mathematics, The Slate University of New York at Potsdam, </p><p>Potsdam, New York 13676 </p><p>Communicated by R. Rosen </p><p>ABSTRACT </p><p>In this work, by employing the concept of vector Lyapunov functions, and the theory </p><p>of differential inequalities, the stability analysis of compartmental systems is initiated in a </p><p>systematic and unified way. Furthermore, an attempt is made to formulate and partially </p><p>resolve the complexity vs. stability problem in open compartmental systems. The recent </p><p>stability results obtained for open chemical systems are applied to open compartmental </p><p>systems in a natural way. As a byproduct of our analysis, we obtain an estimate for </p><p>washout functions. Finally, it has been demonstrated that the stability analysis of </p><p>compartmental systems provides a common conceptual framework for innumerable, </p><p>apparently unrelated systems in biological, medical, and physical sciences. </p><p>1. INTRODUCTION </p><p>The concept of compartmental processes provides suitable means of description for several wide ranges of dynamic processes in biological, physical, and social sciences. A similar fact has been stated by Rosen [22] with respect to morphogenetic and two-factor processes in a systematic and unified form. Furthermore, the scope of compartmental analysis has been well documented in the recent monographs by Jacquez [7] and by Rosen [23], in an excellent paper by Hearon [4], and also in the monograph by Rescigno and Segre [18]. However, a systematic stability analysis of com- partmental systems awaits development. An attempt in this direction has been made by Hearon [4] in connection with the stability of the steady state of linear compartmental systems. For nonlinear compartmental systems, Bellman [2] was the first to formulate the concentration-dependent trans- </p><p>*The research reported herein was supported by SUNY Research Foundation Faculty Fellowship No. 0454-O l-03 1-74-O. </p><p>MATHEMATICAL BIOSCIENCES 30, l-21 (1976) </p><p>0 American Else&x Publishing Company, Inc., 1976 </p><p>1 </p></li><li><p>2 G. S. LADDE </p><p>portation rates and also study the stability of the steady state of closed compartmental systems. In a recent work, Jacquez [7] has used linearized approximations to show the stability of the steady state of nonlinear open compartmental systems. However, these results lack the realism and struc- tural insight of the stability analysis. Lyapunovs second method has been utilized successfully for the stability study of chemical systems [22, 25, 331, controlled biological systems [31, 321, ecosystems [ 121, etc.; however, this method has remained unused for the stability study of compartmental systems, even though it has been recognized that chemical systems [4, 7, 221 and ecosystems [4, 13, 221 can be viewed as compartmental systems. </p><p>Because of the fundamental importance of open compartmental systems for demonstrating analogies between systems at different levels of biological and physical organization, we propose to extend our stability analysis of chemical systems [8] to compartmental systems. This fact will support a very fundamental and important attempt made by Rosen [21, 221 to unify innumerable, structurally diverse dynamic systems in biological and physi- cal sciences. As remarked by Rosen [22], such a unification, with the resultant economy of thought, seems absolutely necessary in these days of explosive growth in scientific knowledge, if the student is not to be hope- lessly submerged under the torrent of specific but specialized investigations relevant to his interest. Indeed, unless expanding knowledge is accompanied by corresponding conceptual economies, a comprehensive understanding of even a small corner of science may become impossible before much longer.. </p><p>Our present stability analysis is the continuation of our stability analysis of chemical systems [8], and it is strongly influenced by the remarkable work of Hearon [4], Jacquez [7], Rescigno and Segre [ 181, and Rosen [21, 221. Furthermore, two of the important classes of dynamical analogies, namely, morphogenetic systems [22, 231 and two-factor systems [19, 221, have been realized as hierarchic compartmental systems. It is important to note that our method of study does not demand either linearization ap- proximations or reduction techniques for analyzing the stability properties of open compartmental systems. </p><p>For linear, nonlinear, and hierarchical open compartmental systems, our stability analysis is analogous to the corresponding stability analysis of linear, nonlinear, and hierarchical open chemical systems [8]. In fact, it is a generalization of the first part of our work [8], so as to include wide varieties of dynamic systems in life and physical sciences. </p><p>The paper is organized as follows: In Sec. 2, we recognize the fact that the transport rate matrix of the </p><p>linear compartmental system is the Metzler matrix 1141 that is used by several economists to obtain strong stability results for economic systems. </p></li><li><p>STABILITY OF COMPARTMENTAL SYSTEMS 3 </p><p>In the present work, it is shown that for the linear compartmental system, the quasidominant diagonal property of the corresponding transport rate matrix is necessary and sufficient condition for stability of the steady state of the linear compartmental system. Based on our study, we conclude that the dominant diagonal property enables us to draw conclusions about the tolerance for increased complexity of stable open compartmental systems. Furthermore, we also recognize a complexity vs. stability problem similar to that in open chemical systems [8] and show that the dominant diagonal property is a suitable mechanism for partially resolving this problem. However, for more details, we refer to [8]. In Sec. 3, nonlinear compartmen- tal systems, and in Sec. 4, hierarchic compartmental systems are briefly formulated, and further to avoid tedium, detailed mathematical analysis, remarks, and conclusions are summarized by reference to the first part of this work [8]. Furthermore. our analysis suggests a notion of quasi-washout function which includes the usual notion of washout function [5], and it gives us an estimate of it in a natural way. In Sec. 5, the scope of our stability analysis of hierarchical compartmental systems is exhibited by identifying classes of dynamical analogies-for example, morphogenetic systems [22, 231 and abstract two-factor systems [19, 22]-as hierarchic compartmental systems. In addition to this, the stability study of several important network problems [19, 20, 221, such as the epigenetic network, simple series network, discrimination network, and learning network, is given as a special case of the stability study of compartmental networks or hierarchical compartmental systems. </p><p>2. LINEAR COMPARTMENTAL SYSTEMS </p><p>Let us consider a compartmental system consisting of n compartments described by a linear differential equation </p><p>i=Ax+a. </p><p>where x E R is a state variable n-vector x =(x1,x2,. .,x,), and T stands for transpose of a vector. For i~1. X, represents the concentration of a labeled observable quantity (e.g., a drug), or the strength of a labeled observable quality (excitatory, or inhibitory, etc.) in the ith compartment, where 1={1,2,..., n}. A = (ati) is the n X n constant observable quantity (or quality) transport rate matrix, and a E R is the constant input rate vector from external sources. In view of the physical significance, the elements of the matrix A must satisfy the conditions: </p><p>ai, < 0, iEI (2) - </p></li><li><p>4 G. S. LADDE </p><p>and </p><p>ay > 0, i#j, iJ E I. (3) </p><p>From (2) and (3) we immediately observe that the constant observable quantity (or quality) transport rate matrix A is a Metzler matrix [14]. Throughout our discussion, we assume that the linear compartmental sys- tem (1) is open. See [7, 18, 231. </p><p>It has been observed [ 1, 21 that the solution x(t, to,x,,) of (1) is non- negative for all tat,, (t,,x,)ER+XR;, where R+=[O,cs), R;={xE R :x, > 0 Vi E I}. Furthermore, the steady state x* of (1) is equal to -A -]a whenever A is nonsingular. Therefore, the stability of x* = -A -a of (1) is equivalent to the stability of x* = 0 of </p><p>.k=Ax. (4) </p><p>For more details, see [8]. By following the argument used in [8], we conclude that the steady state </p><p>x* = 0 of (4) is stable [22] if and only if the Metzler matrix A is a Hicks matrix [ 141. </p><p>The main objective of the present work is to study the effects of the strength of the interactions among the compartments on the stability of the steady state of an open compartmental system under structural perturba- tions. In order to study the effects of the strength of interactions among compartments on the stability of the steady state x* in (1) let us assume that the elements av of the matrix A have the following form: </p><p>(5) </p><p>In (5) eG are elements of the n x n interconnection matrix E =(e,), and they can take on values between zero and one. By eV we represent the strength of connection of the compartment x, with the compartment xi, ranging from disconnection (eV = 0) to full connection (eU = 1). Furthermore, the interconnection matrix E represents the structural changes in the interactions among the compartments in (1). In particular, the shutdown of a pathway from ith compartment to the jth compartment in the compart- mental system is represented by e,, = 0. The disappearance of the pathways between the ith and jth compartments in the system is shown by eU = eji = 0. </p></li><li><p>STABILITY OF COMPARTMENTAL SYSTEMS 5 </p><p>The disconnection or disappearance of the pth compartment in the com- partmental system (1) is represented by eiP = ePj = 0 for all ij E I. If ePi = 0 for all in Z, then the pth compartment is called a pure source compartment [24]; if eip =0 for all in I, then the pth compartment is called a pure sink compartment [24]. If e,P=ePi=O forp#i-l,i+l,pEZ, 2</p></li><li><p>6 G. S. LADDE </p><p>FIG. 5. The disconnection of the FIG. 6. The pth compartment is a </p><p>pth compartment. pure source. </p><p>FIG. 7. The pth compartment is a pure sink. </p><p>FIG. 8. An open catenary compartmental system. </p></li><li><p>STABILITY OF COMPARTMENTAL SYSTEMS </p><p>FIG. 9. A mammillary compartmental system of n + 1 compartments. </p><p>Further note that in (5), (Y; is the rate of excretion from the ith compart- ment (or rate of loss to the environment), and aij is the rate at which material moves from the jth compartment to ith compartment. </p><p>In this section, we investigate the stability analysis of (1) under structural perturbations by means of algebraic conditions. Very recently, such condi- tions were utilized for studying the connective stability property of chemical systems [8] and model ecosystems [lo, 27, 281. Note that we need the concept of fundamental interconnection matrix E [26] and the concept of connective stability [26], which have been utilized in our earlier work, in order to study the stability analysis of (1) under structural perturbations. </p><p>Now we present sufficient conditions for the connective stability of the system (1). Recall that the Metzler matrix is a Hicks matrix if and only if it is a quasidominant diagonal matrix [14], i.e., there exist numbers cr >O, di > 0, i E Z such that </p><p>Iu,I-q i dilaql>a, jEZ. (6) i=l i#j </p><p>Note that the conditions (6) include the usual diagonal dominance: </p><p>as a special case whenever 4. = 1. The condition (7) was utilized by Hearon </p></li><li><p>8 G. S. LADDE </p><p>[4] to deduce the stability of linear systems. As noted in [8], to deduce the connective stability of (1) it is enough to assume that (6) holds for a given fundamental interconnected matrix 6. </p><p>Analogously to linear chemical systems [8], we recognize the complexity vs. stability problem, and conclude that so long as the interaction strength satisfies (6), the stability of (1) -is assured. Thus the dominant diagonal property provides a suitable mechanism for partially resolving this problem. Furthermore, for the linear compartmental system (1) the relation (6) is a necessary and sufficient condition for the stability of the steady state x* of (1). For further details see the first part [8]. </p><p>3. NONLINEAR COMPARTMENTAL SYSTEMS </p><p>Consider the nonlinear time-varying compartmental system described by the system of differential equations </p><p>i=A(t,x)x. (8) </p><p>Here x E D (0, p) is the state variable vector, where D (0, p) is defined by </p><p>D(0,p)={x~R":~x~~0 VIE 1. The n x n transport rate matrix function is A : R, x D(O,p)+R"'. We shall suppose that A is smooth enough to assure the existence and uniqueness of solutions x( t, te, x0) of (8) for all t 2 to, (t,,x,) E R, X D (0,~); further assume that it possesses non-negative solutions whenever the initial state is non-negative. </p><p>It is well known [22] that nonlinear open systems admit a multiplicity of steady states. Therefore, it is essential to study the stability behavior in the vicinity of a given steady state. In view of the discussion in [8], it is enough to assure that x* = 0 is the only steady state of (8) in D (0, p). </p><p>The stability analysis of (8) cannot be formulated in algebraic terms as is possible for the linear constant systems (1). Therefore, we use Lyapunovs second method [I 1, 221 to deduce the stability of the steady state x*=0 of (8). As far as I am aware, this has not yet been done for nonlinear compartmental systems. Our stability analysis confirms the stability analysis conducted by Bellman [2] for closed compartmental systems. </p><p>In order to derive the conditions for connective stability of the compart- mental system (8) we assume that the elements aU(t,x) of the transport rate matrix function A (t,x) have the following form: </p><p>-+i(ttx)+ c ekr(fhki(t3x.)+ IZ ek,(t)+k,(t,x>, i=j a,(r,x) = kEI-(I) kEl(i) </p><p>e,(~)+v(fyx), i#j, </p><p>(10) </p></li><li><p>STABILITY OF COMPARTMENTAL SYSTEMS 9 </p><p>for all (t,x) E R, x D (O,p), where $I~,@~ : R, X D (O,p)dR are functions of (t,x), and V~EZ, Z-(i)={kEZ:k#i, (~~~(f,x) 0, Rki > 0, (Yki > 0, i E I, and </p><p>I > 2 OLki. (12) kEI(i) For the significance as well as some implications of this condition, see [8]. </p><p>By K= (aO) we represent the n x n constant matrix defined by </p><p>(13) </p><p>i#j, </p><p>where g0 are elements of the fundamental interconnection matrix E as defined in [8, 261 and (Y,,N~,R~~ are as defined in (11). </p><p>Consider the linear autonomous auxiliary or comparison system de- scribed by the differential equations </p><p>ti=Ku, (14) </p><p>where ME R:, and 2 is as defined in (13). From (12) and (13), it is obvious that K is a Metzler matrix. </p></li><li><p>10 G. S. LADDE </p><p>In the following, we state the connective stability results (local as well as global) with respect to the system (8). Their proofs follow directly by imitating the proofs of Theorems 1 and 2 in [8]. </p><p>THEOREM I </p><p>Assume that the system (8) satisfies the constraints (11). Then the quasi- dominant diagonal property of the n X n constant Metzler matrix 2 in (14) implies the exponential connective stability of the steady state x* = 0 of the </p><p>system (8). </p><p>THEOREM 2 </p><p>Let the hypotheses of Theorem 1 be satisfied with pi = 00 for every i E I, so that D(O,p)=R. Then the quasidominant diagonal pro...</p></li></ul>

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