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Logical identification of all steady states: The concept of feedback loop characteristic states
Bulletin ofMathemaUcal Bwloqy Vol 55, No 5, pp 973-991, 1993 Printed m Great Britain
0092 8240/9356 0 0 + 0 00 Pergamon Press Ltd © 1993 Socmty for Mathematical Biology
L O G I C A L I D E N T I F I C A T I O N O F ALL STEADY STATES: THE C O N C E P T OF F E E D B A C K L O O P CHARACTERISTIC STATES EL HOUSSTNESNOUSSl Department of Mathematics, University of Kenitra, Morocco RENE THOMAS
Department of Molecular Biology, University of Brussels, Belgium
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 rea~son 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, 10, 1, 20, 2 . . . . . in which 10, 2 0 , . . . 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 identifying all 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 of loop-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 radmal simplification of the identification of the steady states, but in an entirely new view of the relation between feedback loops and steady states.
1. Introduction. Biological regulatory systems often are complex networks comprising several intertwined feedback loops. As amply discussed elsewhere (e.g. Thomas and D'Ari, 1990), the behaviour of such systems is extremely antiintuitive and cannot be solved without an adequate formalization. They can be accurately described by non-linear ordinary differential equations which, however, cannot be solved analytically. Logical approaches (Rashevsky, 1948; Sugita, 1961; Kauffman, 1969; Thomas, 1973; Glass, 1975; Ratner and Tchuraev, 1978) try to extract the essential qualitative features of the dynamics of these systems. Until recently, part of the steady states detected by the differential description could not be 973
974
E. H SNOUSSI A N D R. T H O M A S
identified by logical descriptions. This is because they are located on threshold values, which are not considered explicitly in classical logical descriptions. We call these states "singular", as opposed to the "regular" states, for which none of the variables is located on a threshold. We can now identify all the steady states on logical grounds (Thomas and D'Ari, 1990). In its initial form, this analysis implied a scanning of all possible logical states for steadiness. As the n u m b e r of logical states rapidly increases with the n u m b e r of variables and of logical levels considered, a scanning of all states would have been inconvenient if we did not have ad hoc c o m p u t e r programs. With these programs such a scanning is in no way inconvenient, but it remains esthetically unpleasant. Fortunately, we recently realized (Thomas, 1991 a and b) (1) that one can associate a "loop characteristic state" with each feedback loop, (2) that a m o n g the singular states only those which are loop-characteristic can be steady and (3) that if the characteristic state of a loop is steady, the loop is effective, i.e. it actually generates homeostasis if it is a negative loop and multistationarity if it is a positive loop. This provides not only an enormous simplification of the analysis leading to the identification of steady states, but also a completely new outlook of the relation between feedback loops and steady states. The central aim of this paper is to provide a formal demonstration that a singular state which is not loop characteristic cannot be steady and a state which is loop-characteristic can be steady for appropriate values of the parameters (at least in the subspace of the variables involved in the loop). This formal demonstration, which transforms our conjecture into a theorem, constitutes Section 3 of this paper. The very notions of "singular logical states" and of "loop-characteristic states" are quite new and unfamiliar to most readers. For this reason, it was felt necessary to have the formal Section 3 of this paper preceded by a section which briefly depicts the main steps which have led us to these concepts, using a language more accessible than the mathematical description of Section 3 and in fact a simpler formalism.
2. A Remainder of the Steps Leading to the Concepts of Singular Logical State Loop-characteristic State. 2.1. Use of multilevel logical variables (see Van Ham, 1979). When an element of a system acts on one process only, we give it either of the two logical values 0 (absent) or 1 (present). If an element acts on m processes, we associate m thresholds with the corresponding variable. Let x be the real value of the variable and 10< 20< . . .
IDENTIFICATION OF STEADY STATES
x=0 x=0
if if
x < 10
x= m
if
975
(*)
10"~X<20 x > "O.
However, when we consider one particular process, what we are interested in is whether the real value does or does not exceed the threshold associated with this process. This is why we also use a set of boolean (binary) variables ix, 2x, . . . , mx d e f i n e d b y : ix=0 2x=0
if if
x
lx=l 2X= 1
x>lO x~> 20.
if if
The relation between the real variable x, the multilevel logic variable x and the boolean variables ix, 2 x , . . . , mx is as follows: 0
20 ~ k
X=0 k
m
k,
Y
k
_
x= 2
x=l J
ZX= 0
3(
)
_
)
.
x: real variable
.
x: logical multilevel variable
3(
lx~l 'x, 2x: boolean variables.
k
¥ 2X~ 0
y
~
2X=
1
2.2. Generalized logic using logical parameters (Snoussi, 1989; Snoussi et al. in Thomas and D'Ari, 1990). Logical parameters endow each term of a logical expression with a weight.
Consider the two-variable example
+2 +1 (-"x~_____________A3, ~ ' ) or more compactly: + 1~-" '--'+2
in which variable x positively acts on y above its higher threshold 20 and on itself above its lower threshold 10, and similarly y positively acts on x above its higher threshold 20 and on itself above its lower threshold 10. The piecewise linear differential equations (Glass and Pasternak, 1978) used to describe the evolution of such a network can be written as:
976
E.H. SNOUSSI AND R. THOMAS
dx
~ [ = kaxlx+klz2Y - x dy ~ [ = k2x2x+k221Y-Y; in which ~x, 2y, etc. are our boolean variables and the other symbols refer to real values. Note that the expression k l ~ x + klzZX can take only four (real) values, depending on the values (0 or 1) of the two boolean variables:
0t kll k12 kll =k12
if if if if
~x and 2y both = 0 i x = l but Zy=O ~x = 0 but 2y _~ 1 ~x and 2y both = 1.
We have shown elsewhere (Snoussi, 1989) that the qualitative dynamics of the system can be deduced from the logical equations:
Xy = dx(ktllx+kle2y) d r ( k 2 1 2 x + k 22 1.~ Y)"
(1)
in which the ks are real parameters, ix, 2x, ly and ;y are the same booleans as above, + is the real sum and d x (dy respectively) is the operator which discretizes the real value in the brackets according to the scale on the x (y respectively) axis. In other words, dx(klllx+k122y) is the logical value corresponding to the real value k I ~ix + k~a2y. The logical vector (X, Y) can be viewed as the state toward which the system tends when it is in state (x, y). F o r this reason (X, Y) is called the image of (x, y). We introduce the operators d x, d y , . . , because what interests us here is not so much the real values of the expressions but their logical value, which expresses their location in the discrete scale according to the variable considered. For example X c a n take the values 0, dx(kl 1), dx(kl2 ) o r dx(kl i "~-k12), which we write as 0, K1, K 2 and K12 for short, respectively. Similarly, dy(k2~ + k32 ) is written L12, etc. Thus, equation (1) generates the state table shown in Table 1. The Ks and Ls are called logical parameters. As our parameters K 1 , K 2 and /(12 share the scale of variable x (which in the present case has three values), these parameters can take only the values 0, 1 or 2, with the constraint that K a2 cannot be lower than K 1 or K 2 . Note that the real ks are additive but the logical Ks are not. For example, in the situation: ~ k l0 if there is a basehne, corresponding for example, in genetics, to the constltutwe basic expression of a gene.
IDENTIFICATION O F STEADY STATES kl
k2
l0
20
I
I
I
I
)
y-
x~0
k
)
977
k t + k2 I
k
Y
x=2
x=l
T a b l e 1. x
y 0 1 2 0 1 2 0 1 2
X
Y
0 0 K2 K1 K1 K12 K, K1 K12
0' L2 L2 0 L2 L2 L1 Ll2
with
K1 =d~(k11) /£2 = d~(k,2) K12 = d~(kl i +k12) L 1 = dy (k 21 ) L2 = dy(k22) L12= dy(k21 +k22)
L12
one sees that K 1 and K 2 = 0, but Kt2 = 2; thus, dx(k 11) + dx(k12) may differ from dx(kll -b k12). Each choice of logical parameters leads to a typical qualitative behaviour. If we take, for example:
Kl=K2=L1=L2--1
and
K12=L12=2,
the state table is: Table 2a.
Tabl, 2b.
x
y
X
Y
0
0
0
0
0
1
0
1
x
y
@
X
Y
0
0
0
l
1
1
+
0
2
1
1
0
1
0
1
0
@
1
0
@
1
1
2
2
1
0
1
1
1
2
a
2
1
l
1
1
1
2
2
1
2
0
1
1
2
+
2
1
1
2
2
2
2
2
1 --
+
2
l
The evolution of the system is deduced from a comparison between the state (x, y) and its image (X, Y). The values of the logical functions X, Y (the imageof
978
E.H. SNOUSSI AND R. THOMAS
vector x, y) describe the state toward which the system tends when it is in state (x, y). When X-- x, the variable is steady, when X > x, there is an order for x to increase its logical value and when X < x , an order to decrease its value. Consequently, when there is no order to switch the value of any variable one deals with a logical stable-state solution of X = and Y= y, which are called the logical steady state equations. It is extremely convenient to symbolize an order to increase by a superscript + and an order to decrease by a superscript - on the symbol of the variable. N o superscript is used when the variable is steady, i.e. when x (vs y) equals X (vs Y). For example, instead of writing that the image of state 02 is 11, it is more +
compact to write 02. When a state is stable, it is represented circled. These notations are included in Table 2b (in this case, the inclusion of + and - signs is redundant but convenient); they can also be used to represent the evolution in the phase space as shown in Fig. 1.
20
I
V
6)
®-.
10 --
20
X
10
z0
Figure 1. The state transition graph of system (1).
This graph is called the state transition graph. The arrows represent the +
possible transitions. For example, from state 20 one can proceed to the stable state
or to state 21.
We see that in spite of its structural simplicity, this system has five logical stable states. It was conjectured (Thomas and Richelle, 1988) and later demonstrated (Snoussi, 1989) that each logical stable state corresponds to a stable node in a homologous differential system using step interactions. If instead one uses sufficiently steep sigmoid interactions, the result is essentially the same. As shown in Fig. 2, the homologous differential system has 11 steady states, five of which are stable nodes (cf. linear stability analysis) whose location fits with that of the five logical stable states. But why is it that this logical description only finds part of the total steady states of the system? The reason is simple: the other steady states are located on thresholds, and since so far we had considered the cases x < 0 and x > 0 but not the case x = O, states located on
IDENTIFICATION OF STEADY STATES
6 -
I I
5-
xl£__
,f
I"N
-
y 3
979
I I' I
1
f
,,~
I __I
I I
~llk ' ~ I
-
I j/ ....
I
F - - ii,~ --~TF o
I
")
I
0
1
2
3
,
l
i
4
5
6
X Figure 2. A sigmoid system homologous to system (1):
f ~dx/
yn = kll 4011 - ~x" n n 4- /£12 O]24-yn dy x" y" ] { 2 10~- 1 4-x n 4- k22 0-12 4-yn
k-lX k_2y,
with k l l = 2 , k12=3, k21=3, k22=2, k a = k _ 2 = l 011 and 022~1 012 and 021 = 2 n=20 is the nullcline representative of dx/dt = 0 is the nullcline representative of dy/dt = O. The intersects are the steady states (11) of the system. Linear stability analysis shows that five of them are stable nodes, five are saddle points and one is an unstable node.
thresholds necessarily escaped detection. We will see below that taking this into account, all the steady states detected by tl~e differential method (11 in the present case) can be identified on logical grounds. As a matter of fact, it has become much more convenient to identify them first on logical grounds, and then with the differential method. 2.3. Singular logical states. As mentioned by T h o m a s and D'Ari (1990) and Thomas (1991 a and b), we now ascribe a logical value to the real thresholds themselves. Let 10 be the first threshold of variable x. W h e n x takes the value 10 we introduce the logical symbol a0 and write x = 10. If variable x has two thresholds 10 and 20, we thus have five logical values: x = 0, x = 10, x = 1, x = 20
980
E . H . SNOUSSI A N D R. T H O M A S
and x = 2. We now have regular states, in which no logical variable is located on a threshold, and singular states, in which one or more variable has a threshold value. If the thresholds of variable j in Equation i are symbolized 10, 2 0 , . . . , 'nO, we write that its order pij= 1, 2 , . . . , m. How singular states can be fed into the logical equations, in order to get the value of their image, is described in the Appendix. Suffice to remark here that in the case of a singular state the resulting image of one or more variable can be an interval rather than a point, and in this case the compatibility with the steady state equation X = x means that the threshold value x is comprised in the interval of X. One reason for rejecting this section to the Appendix is that the theorem described in the second part allows us in practice to completely avoid this type of procedure when we check whether or not a state is steady. Nevertheless, this procedure had to be described somewhere in order to show that singular states indeed can be fed into our logical equations. 2.4. Loop-characteristic states. The characteristic state of a feedback loop is defined as the state for which each variable of the loop-~ is located at the threshold value, above which it is active in the loop considered. For example, in system (1) there is a loop between variables x and y, both acting above their second threshold 20; thus, the characteristic state of this loop is xy = 2020. Needless to say, when a loop involves only part of the variables of the system, the characteristic state is considered in the subspace of the relevant variables only. For example, in system (1) there is a loop of variable x on itself and a loop of variable y on itself, both acting above their lower threshold 10. The corresponding characteristic states are 1 0 - in the subspace of variable x, and - 1 0 in the subspace of variable y. Moreover, as these loops are disjointed we also have to consider their reunion, whose characteristic state is 10~0, this time in the whole space of the system. It has turned out for some time that among the singular states, only those which are loop-characteristic can be steady. More specifically, a singular state which is not loop-characteristic cannot be steady whatever the parameter values, and a state which is loop-characteristic can be steady (at least in the subspace of the variables involved in the loop) for appropriate values of the parameters (Thomas, 1991 a and b). This was only a conjecture (based on the analysis of m a n y specific situations) until it recently became a theorem: see Section 3 of this work. 2.5. Logical identification of all steady states. How do we now identify the steady states of a system in practice? As regards the regular states, one just has to compare the state with its image: if they are identical, the state is steady. As Or, more generally, of the reunion of disjointed loops, I.e. of loops which share no variable.
IDENTIFICATION OF STEADY STATES
981
regards the singular states, in view of the theorem just mentioned, we consider only those which are loop-characteristic. As described in a more rigorous way (and with a more elaborate formalism) in Section 3, we reason that in the vicinity of a loop-characteristic state, the only operative interactions are those which constitute the loop itself. The characteristic state of an n-variable loop has 2" adjacent regular states, but it suffices to consider those two adjacent states whose images are maximal and minimal. To find these two states, one simply gives to each variable the values just above and just below the threshold considered if the variable acts positively, and the values just below and just above the threshold if the variable acts negatively.
x~y~
+1
z
For example, for the loop t [, the characteristic state is 201030 -3 (whose orders, p, are 2, 1 and 3, respectively) and the two adjacent states whose images are maximal and minimal are 212 and 103, respectively. Let AIB~C 1 and A2B2C 2 be the images of these states. In order to be consistent with the steady state equations, the orders of the relevant thresholds must be comprised in the interval of the images; more precisely, in the present case one must have A 2 < 2 0 < A 1 , B 2 < 10
y
X
Y
1
1
K1
L2
2
2
K12
L12
(extracted from Table 1). As one can see, state 2020 will be steady iffthe images of the adjacent tates encompass 2020. More specifically, one must have K 1 and L 2 < 20; K12 and L12 > 20. As our parameters can take only the values 0, 1 or 2 (like the corresponding variable), this means that the conditions for 2020 to be steady are: K 1 , Z 2 = 0 or 1 K12, L 1 2 = 2 .
(2) The characteristic state of loop +x~,) 1 is 1 0 - ; we thus have to consider the various possible values of y. Each of states 100, ~01 and ~02 will thus be
982
E.H. SNOUSSI AND R. THOMAS
considered for stationarity, and for each of them we will consider the two adjacent states and their images. For state ~00 the table is:
For state 101 the table is:
For state 102 the table is:
x
y
X
Y
x
y
X
Y
x
y
0 1
0 0
0 K1
0 0
0 1
l l
0 K1
L2 L2
0 1
This state is steady iff: K 1 = 1 or 2.
This state is steady iff: L 2= l and K~= 1 or 2.
X
Y
2
K2
L2
2
Klz
L2
This state is steady iff: L 2=2, K 2 = 0
and K12 = 1 or 2. N o t e t h a t t h e c o n s t r a i n t s a r e c o n t r a d i c t o r y for ~ 0 ~
(Klz--~-1) a n d 1 0 Q ~
( K 1 = 2); i n c o n t r a s t , e a c h o f t h e s e s t a t e s c a n c o e x i s t w i t h Q 0 0 ) . (3) F o r t h e c h a r a c t e r i s t i c s t a t e - 10, w e h a v e t o c o n s i d e r stateVs 010, 110 a n d 220: For state 010 the table is:
For state 110 the table is:
For state 210 the table is:
x
y
X
Y
x
y
X
Y
x
y
X
Y
0 0
0 1
0 0
0 L2
1 1
0 1
K1 K1
0 Lz
2 2
0 1
Kx Ka
L12
The condition of steadiness is: L 2 = 1 or 2.
The conditions are: L 2 = l or 2 and K1=1.
The steady states @
and @
Li
The conditions are: L t = 0 , L12=1 or 2, K1=2.
cannot coexist since @
requires
K 1 = 1 and (210) requires K 1= 2, but either of these states can coexist with
@
(4) As the loops +x ) l a n d
+y)1
are disjointed we have also to
consider their reunion, whose characteristic state is 1010. The conditions of stationarity, given by the table: x
y
X
Y
0
0
1
1
0 K1
L2
0
are simply K 1 and L 2 = 1 or 2. From this, one can see that for the set of parameters chosen above the system admits, in addition to the five stable states 00, 01, 10, 11 and 22 already mentioned, the six singular steady states 2020, 100, 101,010, 110 and 1010. A look at Fig. 2 shows that these logical steady states exactly fit with the steady states of the differential description.
I D E N T I F I C A T I O N O F STEADY STATES
983
Three remarks: (1) All this procedure is now completely informatized (Thieffry et al., 1992). Given the logical parameters a program returns to the logical steady states (and their nature). However, the most interesting approach consists of asking the parametric constraints for any characteristic state (or any combination of characteristic states) to be steady. (2) Given the logical parameters, one can easily find consistent real parameters and inject them into a set of differential equations (with step or sigmoid interactions). As described in Thomas and D'Ari (1990), the relation between the logical and real parameters is as follows: For the stepwise linear equation: 2 = k(x > toO)- k _ x, the logical constraint K > m simply means that k > m0. If one uses sigmoid interactions, the relation is k >amO, in which the value of a depends on the steepness of the sigmoid (a = 1 for a step function, a = 2 for a smooth sigmoid with a Hill number of 2). In fact, it is in this way that we have found the real parameter values to be used in Fig. 2. (3) Not only the location but also the nature of the steady state can be predicted on logical grounds. (4) For pedagogic reasons, we have reasoned here in terms of homogeneous logical equations. In practice, we always consider the possibility that a gene can have a residual activity even in less favourable conditions. Formally, this amounts to having an independent term in each logical equation: X=dx(klo + kllmx + ...). As already mentioned, all these operations can now be performed by appropriate programs (Thieffry et al., 1992).
3. Formal Description. 3.1. Notations and definitions. differential equations: dXi dt
We consider the system of piecewise linear
- kio+ ~ kijS~zJ(Xi/Oij)-kiXi, i=1
in which k i j ) O , ki>0, a o ~ { + ; - } S~'J(X/Oij)e{O, 1} is defined by:
i = 1 , 2 , . . . n;
(2)
defines the sign of the interaction and (chj--+
and
Xj>Oij
~ii---
and
Xj
=
984
E . H . SNOUSSI A N D R. THOMAS
The thresholds Ozj, corresponding to a given variable Xj, are different and are arranged in ascending order on the axis Xj, thus, to each threshold Ozj corresponds an order noted p~. The interaction graph which describes system (1) is an oriented, signed and weighted graph defined as follows. The n vertices are numbered from 1 to n and an arrow (j, i, cqj, Pij) expresses the fact that the ith equation depends on the variable Xj (i.e. kij ~ 0) acting above the threshold 0~ of the order p~ and eij is the sign of the interaction. The discrete m a p p i n g used in Section 2 to write the state formally is written as:
~o,(d(X))=di(Ooi(X)),
i= 1, 2 , . . . n,
(3)
with (I)i(X) = 1/k~[kio + ~ : l k~jS~'J(Xj/O~j)], i= 1, 2 , . . . n and d is the operator of space discretization defined by: d(X,,
X2 ....
X,) = (d,(Xx),
d2(X2),
. . .
d,(X,)),
with di(Xi)=r if and only if Xi is comprised between the two consecutive thresholds of respective order prz and p~+ 1,z. In a preceding paper (Snoussi, 1989) it was shown that the discrete m a p p i n g q) permits localizing all those stable nodes which are not located on a threshold. In the following we will show how to obtain all the steady states of system (1) using only the m a p p i n g q~. 3.2. Singular steady states. Definition 1. A state A is said to be singular if at least one of its components takes a threshold value. We note A =A(L J) in order to express that:
Viii, 3jeJ such that A i = Oji. A state not located on any threshold is called regular.
Definition 2. Let B = ( i 1 - - - ~ i 2 - - * • • • ir~il), a loop in the interaction graph. A singular state A is said to be characteristic of a loop B if: Vm~{1, 2 , . . . r}, Ai,=O,m+,,i m (m + 1 is the calculated m o d u l o r). In other words, A = A (/, J) is characteristic of a loop ifJis a permutation of/. More generally, a singular state is characteristic of a reunion of disjointed loops if it is characteristic of each loop. In Equations (2) and (3), the applications d, S + and S - had been defined only for the regular states. If one considers the singular states, it is natural to take:
IDENTIFICATION OF STEADY STATES
985
d,(Oj~)= Oj~, S~',(Ou/Oo)= ]0, 1 [ c and d,([a i, bi])= Ida(a,), d~(b~)].
Definition 3. A singular state A =A(L J) is said to be steady if for all XeN", such that d(X)=A, we have: ViCL A~= (Pi(A); VieI, ~,(X) is an interval ]a, b[; VieI, if Xj= Oij then Oijc]a, b[. This definition is justified by the fact that when Xj = O~j, S~',(Xj/Ou) is the interval ]0, 1[ and the differential equations (1) become equations of differential appartenance with steady state equations X~E @~(X) for a singular state. THEOREM 1. Any singular steady state is the characteristic state of a reunion of
disjointed loops. In other words, singular states which are not characteristic of a loop cannot be steady whatever the values of the parameters. Proof. Let A = A (/, J), a singular state which is not characteristic of a reunion of disjointed loops. By definition we have:
V,~L 3j~J: Ai=Oji; J is not a permutation of I. Then there exists, necessarily, an element ieI, which does not belong to J and thus:
3m6J such that A i = Omi with i~J. If X is such that d(X) = A, we have:
kio
1
Oi(X) = k i ~- ki
j:l
kuS~"(XjOij),
since i~J, then Xj• O,j for allje{1, 2 , . . . n} and Oi(X) is not an interval. This expresses that A is not a steady state. 3.3. Conditions ofstationarity of singular states. In this paragraph, we give a practical method to recognize whether a singular state characteristic of a reunion of loops is, or is not, steady. Without loss of generality, we consider the state: A = ( 0 2 1 , 0 3 2 , . . . Or,r_1, Ol,r, A r + l , . . .
A,),
986
E.H. SNOUSSI AND R. THOMAS
which is characteristic of the loop 1~2--, -.. --*r. A necessary condition for A to be steady is that:
n, Ai=qoi(A ).
for i = r + l , . . ,
Let pi be the order of 0~+ ~,~for i = 1, 2 , . . . r and X~N" such that d(X)=A. We have for i = 1, 2 , . . . r:
kio ki i- ~ S ..... 1 i ¢,(xt = ~ + ~ ~(xi_~/%_O+ ~ i=1 k,S~'J(X/%) jg:i-1
1 ~
kijSa'J(Xj/Oij )"
~- k / j = r + l
For j ¢ { 1 , 2 , . . . r } , Xj=Oj+I, j and when j # i - 1 , Xj#Oo, thus for j e {1, 2 , . . . r} a n d j # i - 1 we have S~'J(Xj/O,j) taking a value 0 or 1. F o r j e {r + 1 . . . . n}, Xj does not have a threshold value and S~'J(X/Oij) takes a value 0 or 1. Let:
K~ - k,o
1 ~
ki -t- ki
k,~S~,4X/O,) '
j=l
j¢i-1
we thus have, for ie{1, 2, . . . r}: ki i
• ~(X)=K~ + ~
1
S='-'.'(X,_l/Oi,i_l),
since X,_ 1 =Oi.i_l, S='-*.'(Xi_l/Oi,i_l)=]O, 1[ and: ¢~(X)=]Ki, G +
ki,i - 1 [
k~
"
The singular state A = d(X) is steady iff: for i~r+ l, q~i(A)=Ai; for i<~r, ~(X) is an interval which contains Oi+ 1,,. The first condition is given directly from the mapping q~ and for the second condition we must have:
K~
ki '
i.e.
IDENTIFICATION OF STEADY STATES
(4)
di(Ki)
987
}"
Let us calculate di(Ki) and di(K i + k~,i_ 1/ki) using only the regular states A - and A + defined as follows: A i- = A i for i e { r + l . . . . n} Ai- _ ~ Pi~ l - {,Pi
if o~i,i+ l = --{- for ie{1, 2 , . . . r}, if cq,i+ 1 = --
and Ai + = A i for i ¢ { r + 1. . . . n} Ai + =~pt t Pi-1
if if
-I- for ie{1, 2 , . . . r}. ai,i + l = --
O~i,i+ l =
Let X - and X + be such that d ( X - ) = A ie{1, 2 , . . . r}: Oi(X-
and d ( X + ) = A +, we have, for
) = Ki, i.e. di(Ki) = d,(@~(X-))
= ~o,(d(X
))
= q0~(A - ),
and ~ ( X +) = G + ~ki i 1 i.e. di (Ki + k',i--,'~ k i /I =di(O'(X+ )) = ~ ( d ( X +)) = ~Pi(A + ).
Condition (3) becomes: qoi(A - )< pi <<.cPi(A +). We thus have the following result: THEOREM 2. A singular state A = A(I, J) which is characteristic of a reunion of disjointed loops is a steady state if and only if: q)i(A~) =A}, ~= + O r - , j b r iq~I; goi(A- ) < pi <~qgi(A + ) for i6I. This theorem gives us a simple method to recognize if a singular state is steady or not using only the discrete mapping (? or equivalently the state table.
988
E.H. SNOUSSI AND R. THOMAS
Example.
Let the interaction graph be: +1
<_-.-
2
%/ /
- 1
/ - 3
-2
4<
3 +2
Let
d=021, 042, 1,014. +1
-3
-1
This singular state is characteristic of the loop 1 ~ 2 ~ 4 ~ 1 with I = {1, 2, 4} and J = {2, 4, 1}. We have A - =0311, A ÷ = 1210 and:
(klo k,2 k2o k3o k!o'~, e(A-)=dkk, + kx'ke'G'k4J klO
k12
~(A +)=d\ k, + V( +
k~ k2o k2~ Go k4o k42) k I ' k2 q- k 2 ' k 3 ' k 4 q- k 4/]"
A is steady iff:
dlt(kl° -~
kll
d(k2°~
)
)"
4. Summary and Conclusions. Typically, a model is built specifically in order to account for a certain n u m b e r of pre-existing experimental data. However, except for especially simple cases, a model implies a n u m b e r of additional predictions, most of which usually remain implicit, often even undetected. This is especially the case in the field of biological regulation in which the frequent presence of multiple intertwined feedback loops renders predictions about the dynamics far from obvious.
I D E N T I F I C A T I O N O F STEADY STATES
989
In the analysis of the dynamics of such complex systems, a crucial element is the identification of the number, nature and location of the steady states, and also of the domains in the parameter space in which the system behaves in qualitatively different ways. Our previous work had introduced the notion of logical parameters (Snoussi, 1989) and of singular logical states (Thomas and D'Ari, 1990; Thomas, 1991 a and b), which permit the identification of all the steady states on a logical basis. The concept of loop-specific state (Thomas, 1991 a and b) originates from the observation that when a feedback loop is effective (i.e. actually generates homeostasis or multistationarity), there is a steady state at the intersect of the thresholds of the loop. It was soon realized that among the singular states only those which are loop-specific can be steady, and this is formally demonstrated in Section 3 of this paper. From the practical viewpoints the identification of the steady states is greatly simplified, because instead of being obliged to scan all the (often very many) singular states for steadiness one has to consider only those (often very few) which are loop-specific. In practice, it is extremely easy to identify loops and their characteristic state, using a program if required by the number of variables (Thieffry et al., 1991 ), and the derivation of all steady states is simple. From a more conceptual viewpoint: (1) negative feedback loops can generate homeostasis, and the criterion for efficient homeostasis is the presence of a steady state--more precisely, a focus--located at thresholds of the variables involved in the loop; (2) positive feedback loops can generate multistationarity,t and the criterion is the appearance of a steady state--at the thresholds of the variables involved in the loop; this saddle point is usually on the border between two or more basins (characterized each by a steady state), dependng on the number of positive feedback loops. The recent progress considerably clarifies the situation, if only by introducing the generalization that the criterion for the effectiveness of a loop is that its characteristic state be steady. According to these views, in the absence of an effective positive loop a system can only have a single steady state. This means that if this system has several negative loops, not more than one of these loops (or more generally one reunion of disjointed loops) can be effective; which one depends on the parameter values. We expect, and systematically find indeed, that in the absence of a positive loop the conditions of effectiveness of negative loops are mutually exclusive (except if they share no variable). If there is one effective positive loop (i.e. there is one positive loop-specific t W h a t IS still lacking is a formal demonstration that a positive loop is a necessary condition for multistationarity and a negative loop a necessary condition for homeostasis.
990
E.H. SNOUSSI AND R. THOMAS
steady state) the system has three steady states. The loop-characteristic state, which is a saddle point, is on a separatrix which partly or completely separates the space of the variables of the loop into two basins, each characterized by an additional steady state. If in addition to the positive loop there are negative loops, these additional steady states (or one of them) may be homeostatic (e.g. a focus located on the thresholds of the negative loop). How can one have many steady states? The simple recipe is to have several positive feedback loops. If there are only effective disjointed positive feedback loops, m, such loops generate 3" steady states, 2m of which are stable (thus, seven genes subject to a positive autoregulation can account for 27 = 128 stable cell types). Additional interactions usually reduce the number of steady states (Thomas and Richelle, 1988), except if they introduce additional positive loops, as in our example (1).
APPENDIX In order to be able to feed singular states into our logical equations, we have to give a meaning to the binary variables ix, 2x . . . . when x takes a real value located on a threshold. It is natural to extend the definitions in the following way:
ix=
0,1[ if x=lO, 2x= if x> ~O
0,1[ i f x = 2 0 , if x> 20
where ]0, 1[ is the real interval. The logical equataons already used before including the singular states have the general form:
A singular state is said to be steady if for the variables x not on a threshold we have X=x, and for variables on a threshold value 0, the right member of equation (1) is an interval which encompasses the real value 0. Let us illustrate this definition with example (1):
=dx(klllX+k122y) ~cy-d~(k21 x+k22 Y) --
2
1
"
How can we calculate the image of a singular state--say xy = 101 with the parameter values chosen above? For x = 10 and y = l we have ~x=]0, t [ 2 x = 0 , l y = l and 2y=0. Introducing these values m the logical equations, we obtain:
Y=dr(k22)=L2=l, and since y = 1, we have Y= y.
klllX+k122y=]O, k i l l ,
IDENTIFICATION OF STEADY STATES
991
and since K 1= 1, 10
and
X - d~(]0, k11[) =]0, Kt[ Y = L 2.
Since for this state x = 10 and y = 1 the state is steady if 0 < ~0 < K 1 (or K 1~>1) and L 2 = 1. For state x y = 1020, we have: ~x=]0, l [ 2 x = 0 , ~y= 1, 2y=]0, 1[, thus Y=dy(k22), which is not an interval whatever the value of k22. Thus, 1020 is not steady for any values of the parameters, as expected for a singular state which is not loop-characteristic.
LITERATURE Glass, L. 1975. J. theor. Biol. 54, 85-107. Glass, L. and J. S. Pasternak. 1978. Bull. math. Biol. 40, 27-44. Kauffman, S. A. 1969. J. Theor. Biol. 22, 437-467. Rashevsky, N. 1948. Mathematical Biophysics. Chicago: University of Chicago Press. Ratner, V. A. and R. N. Tchuraev. 1978. Prog. theor. Biol. 5, 81 127. Snoussi, E. H. 1989. Dynamics Stability Sys. 4, 18%207. Snoussi, E. H., R. Thomas and R. D'Ari. 1990. In: Biological Feedback. R. Thomas and R. D'Ari (Editors), Ch. 7. Boca Raton, Florida: CRC Press. Sugita, M. 1961. J. theor. Biol. 1,415-430. Thieffry, D., M. Colet and R. Thomas. 1991. Proc. 8th International Conference on Mathematical and Computer Modelling. Oxford: Pergamon Press, in press. Thomas, R. 1973. J. theor. Biol. 42, 563-585. Thomas, R. 1991a. T. theor. Biol., in press. Thomas, R. 1991b. Lecture Notes Bmmath., in press. Thomas, R. and R. D'Ari. 1990. Biological Feedback. Boca Raton, Florida: CRC Press. Thomas, R. and J. Richelle. 1988. Discr. appl. Math. 9, 381-396. Van Ham, P. 1979. Lecture Notes Biomath. 29, 326 343.
R e c e i v e d 3 N o v e m b e r 1991 R e v i s e d 31 J u l y 1992
0092 8240/9356 0 0 + 0 00 Pergamon Press Ltd © 1993 Socmty for Mathematical Biology
L O G I C A L I D E N T I F I C A T I O N O F ALL STEADY STATES: THE C O N C E P T OF F E E D B A C K L O O P CHARACTERISTIC STATES EL HOUSSTNESNOUSSl Department of Mathematics, University of Kenitra, Morocco RENE THOMAS
Department of Molecular Biology, University of Brussels, Belgium
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 rea~son 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, 10, 1, 20, 2 . . . . . in which 10, 2 0 , . . . 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 identifying all 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 of loop-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 radmal simplification of the identification of the steady states, but in an entirely new view of the relation between feedback loops and steady states.
1. Introduction. Biological regulatory systems often are complex networks comprising several intertwined feedback loops. As amply discussed elsewhere (e.g. Thomas and D'Ari, 1990), the behaviour of such systems is extremely antiintuitive and cannot be solved without an adequate formalization. They can be accurately described by non-linear ordinary differential equations which, however, cannot be solved analytically. Logical approaches (Rashevsky, 1948; Sugita, 1961; Kauffman, 1969; Thomas, 1973; Glass, 1975; Ratner and Tchuraev, 1978) try to extract the essential qualitative features of the dynamics of these systems. Until recently, part of the steady states detected by the differential description could not be 973
974
E. H SNOUSSI A N D R. T H O M A S
identified by logical descriptions. This is because they are located on threshold values, which are not considered explicitly in classical logical descriptions. We call these states "singular", as opposed to the "regular" states, for which none of the variables is located on a threshold. We can now identify all the steady states on logical grounds (Thomas and D'Ari, 1990). In its initial form, this analysis implied a scanning of all possible logical states for steadiness. As the n u m b e r of logical states rapidly increases with the n u m b e r of variables and of logical levels considered, a scanning of all states would have been inconvenient if we did not have ad hoc c o m p u t e r programs. With these programs such a scanning is in no way inconvenient, but it remains esthetically unpleasant. Fortunately, we recently realized (Thomas, 1991 a and b) (1) that one can associate a "loop characteristic state" with each feedback loop, (2) that a m o n g the singular states only those which are loop-characteristic can be steady and (3) that if the characteristic state of a loop is steady, the loop is effective, i.e. it actually generates homeostasis if it is a negative loop and multistationarity if it is a positive loop. This provides not only an enormous simplification of the analysis leading to the identification of steady states, but also a completely new outlook of the relation between feedback loops and steady states. The central aim of this paper is to provide a formal demonstration that a singular state which is not loop characteristic cannot be steady and a state which is loop-characteristic can be steady for appropriate values of the parameters (at least in the subspace of the variables involved in the loop). This formal demonstration, which transforms our conjecture into a theorem, constitutes Section 3 of this paper. The very notions of "singular logical states" and of "loop-characteristic states" are quite new and unfamiliar to most readers. For this reason, it was felt necessary to have the formal Section 3 of this paper preceded by a section which briefly depicts the main steps which have led us to these concepts, using a language more accessible than the mathematical description of Section 3 and in fact a simpler formalism.
2. A Remainder of the Steps Leading to the Concepts of Singular Logical State Loop-characteristic State. 2.1. Use of multilevel logical variables (see Van Ham, 1979). When an element of a system acts on one process only, we give it either of the two logical values 0 (absent) or 1 (present). If an element acts on m processes, we associate m thresholds with the corresponding variable. Let x be the real value of the variable and 10< 20< . . .
IDENTIFICATION OF STEADY STATES
x=0 x=0
if if
x < 10
x= m
if
975
(*)
10"~X<20 x > "O.
However, when we consider one particular process, what we are interested in is whether the real value does or does not exceed the threshold associated with this process. This is why we also use a set of boolean (binary) variables ix, 2x, . . . , mx d e f i n e d b y : ix=0 2x=0
if if
x
lx=l 2X= 1
x>lO x~> 20.
if if
The relation between the real variable x, the multilevel logic variable x and the boolean variables ix, 2 x , . . . , mx is as follows: 0
20 ~ k
X=0 k
m
k,
Y
k
_
x= 2
x=l J
ZX= 0
3(
)
_
)
.
x: real variable
.
x: logical multilevel variable
3(
lx~l 'x, 2x: boolean variables.
k
¥ 2X~ 0
y
~
2X=
1
2.2. Generalized logic using logical parameters (Snoussi, 1989; Snoussi et al. in Thomas and D'Ari, 1990). Logical parameters endow each term of a logical expression with a weight.
Consider the two-variable example
+2 +1 (-"x~_____________A3, ~ ' ) or more compactly: + 1~-" '--'+2
in which variable x positively acts on y above its higher threshold 20 and on itself above its lower threshold 10, and similarly y positively acts on x above its higher threshold 20 and on itself above its lower threshold 10. The piecewise linear differential equations (Glass and Pasternak, 1978) used to describe the evolution of such a network can be written as:
976
E.H. SNOUSSI AND R. THOMAS
dx
~ [ = kaxlx+klz2Y - x dy ~ [ = k2x2x+k221Y-Y; in which ~x, 2y, etc. are our boolean variables and the other symbols refer to real values. Note that the expression k l ~ x + klzZX can take only four (real) values, depending on the values (0 or 1) of the two boolean variables:
0t kll k12 kll =k12
if if if if
~x and 2y both = 0 i x = l but Zy=O ~x = 0 but 2y _~ 1 ~x and 2y both = 1.
We have shown elsewhere (Snoussi, 1989) that the qualitative dynamics of the system can be deduced from the logical equations:
Xy = dx(ktllx+kle2y) d r ( k 2 1 2 x + k 22 1.~ Y)"
(1)
in which the ks are real parameters, ix, 2x, ly and ;y are the same booleans as above, + is the real sum and d x (dy respectively) is the operator which discretizes the real value in the brackets according to the scale on the x (y respectively) axis. In other words, dx(klllx+k122y) is the logical value corresponding to the real value k I ~ix + k~a2y. The logical vector (X, Y) can be viewed as the state toward which the system tends when it is in state (x, y). F o r this reason (X, Y) is called the image of (x, y). We introduce the operators d x, d y , . . , because what interests us here is not so much the real values of the expressions but their logical value, which expresses their location in the discrete scale according to the variable considered. For example X c a n take the values 0, dx(kl 1), dx(kl2 ) o r dx(kl i "~-k12), which we write as 0, K1, K 2 and K12 for short, respectively. Similarly, dy(k2~ + k32 ) is written L12, etc. Thus, equation (1) generates the state table shown in Table 1. The Ks and Ls are called logical parameters. As our parameters K 1 , K 2 and /(12 share the scale of variable x (which in the present case has three values), these parameters can take only the values 0, 1 or 2, with the constraint that K a2 cannot be lower than K 1 or K 2 . Note that the real ks are additive but the logical Ks are not. For example, in the situation: ~ k l0 if there is a basehne, corresponding for example, in genetics, to the constltutwe basic expression of a gene.
IDENTIFICATION O F STEADY STATES kl
k2
l0
20
I
I
I
I
)
y-
x~0
k
)
977
k t + k2 I
k
Y
x=2
x=l
T a b l e 1. x
y 0 1 2 0 1 2 0 1 2
X
Y
0 0 K2 K1 K1 K12 K, K1 K12
0' L2 L2 0 L2 L2 L1 Ll2
with
K1 =d~(k11) /£2 = d~(k,2) K12 = d~(kl i +k12) L 1 = dy (k 21 ) L2 = dy(k22) L12= dy(k21 +k22)
L12
one sees that K 1 and K 2 = 0, but Kt2 = 2; thus, dx(k 11) + dx(k12) may differ from dx(kll -b k12). Each choice of logical parameters leads to a typical qualitative behaviour. If we take, for example:
Kl=K2=L1=L2--1
and
K12=L12=2,
the state table is: Table 2a.
Tabl, 2b.
x
y
X
Y
0
0
0
0
0
1
0
1
x
y
@
X
Y
0
0
0
l
1
1
+
0
2
1
1
0
1
0
1
0
@
1
0
@
1
1
2
2
1
0
1
1
1
2
a
2
1
l
1
1
1
2
2
1
2
0
1
1
2
+
2
1
1
2
2
2
2
2
1 --
+
2
l
The evolution of the system is deduced from a comparison between the state (x, y) and its image (X, Y). The values of the logical functions X, Y (the imageof
978
E.H. SNOUSSI AND R. THOMAS
vector x, y) describe the state toward which the system tends when it is in state (x, y). When X-- x, the variable is steady, when X > x, there is an order for x to increase its logical value and when X < x , an order to decrease its value. Consequently, when there is no order to switch the value of any variable one deals with a logical stable-state solution of X = and Y= y, which are called the logical steady state equations. It is extremely convenient to symbolize an order to increase by a superscript + and an order to decrease by a superscript - on the symbol of the variable. N o superscript is used when the variable is steady, i.e. when x (vs y) equals X (vs Y). For example, instead of writing that the image of state 02 is 11, it is more +
compact to write 02. When a state is stable, it is represented circled. These notations are included in Table 2b (in this case, the inclusion of + and - signs is redundant but convenient); they can also be used to represent the evolution in the phase space as shown in Fig. 1.
20
I
V
6)
®-.
10 --
20
X
10
z0
Figure 1. The state transition graph of system (1).
This graph is called the state transition graph. The arrows represent the +
possible transitions. For example, from state 20 one can proceed to the stable state
or to state 21.
We see that in spite of its structural simplicity, this system has five logical stable states. It was conjectured (Thomas and Richelle, 1988) and later demonstrated (Snoussi, 1989) that each logical stable state corresponds to a stable node in a homologous differential system using step interactions. If instead one uses sufficiently steep sigmoid interactions, the result is essentially the same. As shown in Fig. 2, the homologous differential system has 11 steady states, five of which are stable nodes (cf. linear stability analysis) whose location fits with that of the five logical stable states. But why is it that this logical description only finds part of the total steady states of the system? The reason is simple: the other steady states are located on thresholds, and since so far we had considered the cases x < 0 and x > 0 but not the case x = O, states located on
IDENTIFICATION OF STEADY STATES
6 -
I I
5-
xl£__
,f
I"N
-
y 3
979
I I' I
1
f
,,~
I __I
I I
~llk ' ~ I
-
I j/ ....
I
F - - ii,~ --~TF o
I
")
I
0
1
2
3
,
l
i
4
5
6
X Figure 2. A sigmoid system homologous to system (1):
f ~dx/
yn = kll 4011 - ~x" n n 4- /£12 O]24-yn dy x" y" ] { 2 10~- 1 4-x n 4- k22 0-12 4-yn
k-lX k_2y,
with k l l = 2 , k12=3, k21=3, k22=2, k a = k _ 2 = l 011 and 022~1 012 and 021 = 2 n=20 is the nullcline representative of dx/dt = 0 is the nullcline representative of dy/dt = O. The intersects are the steady states (11) of the system. Linear stability analysis shows that five of them are stable nodes, five are saddle points and one is an unstable node.
thresholds necessarily escaped detection. We will see below that taking this into account, all the steady states detected by tl~e differential method (11 in the present case) can be identified on logical grounds. As a matter of fact, it has become much more convenient to identify them first on logical grounds, and then with the differential method. 2.3. Singular logical states. As mentioned by T h o m a s and D'Ari (1990) and Thomas (1991 a and b), we now ascribe a logical value to the real thresholds themselves. Let 10 be the first threshold of variable x. W h e n x takes the value 10 we introduce the logical symbol a0 and write x = 10. If variable x has two thresholds 10 and 20, we thus have five logical values: x = 0, x = 10, x = 1, x = 20
980
E . H . SNOUSSI A N D R. T H O M A S
and x = 2. We now have regular states, in which no logical variable is located on a threshold, and singular states, in which one or more variable has a threshold value. If the thresholds of variable j in Equation i are symbolized 10, 2 0 , . . . , 'nO, we write that its order pij= 1, 2 , . . . , m. How singular states can be fed into the logical equations, in order to get the value of their image, is described in the Appendix. Suffice to remark here that in the case of a singular state the resulting image of one or more variable can be an interval rather than a point, and in this case the compatibility with the steady state equation X = x means that the threshold value x is comprised in the interval of X. One reason for rejecting this section to the Appendix is that the theorem described in the second part allows us in practice to completely avoid this type of procedure when we check whether or not a state is steady. Nevertheless, this procedure had to be described somewhere in order to show that singular states indeed can be fed into our logical equations. 2.4. Loop-characteristic states. The characteristic state of a feedback loop is defined as the state for which each variable of the loop-~ is located at the threshold value, above which it is active in the loop considered. For example, in system (1) there is a loop between variables x and y, both acting above their second threshold 20; thus, the characteristic state of this loop is xy = 2020. Needless to say, when a loop involves only part of the variables of the system, the characteristic state is considered in the subspace of the relevant variables only. For example, in system (1) there is a loop of variable x on itself and a loop of variable y on itself, both acting above their lower threshold 10. The corresponding characteristic states are 1 0 - in the subspace of variable x, and - 1 0 in the subspace of variable y. Moreover, as these loops are disjointed we also have to consider their reunion, whose characteristic state is 10~0, this time in the whole space of the system. It has turned out for some time that among the singular states, only those which are loop-characteristic can be steady. More specifically, a singular state which is not loop-characteristic cannot be steady whatever the parameter values, and a state which is loop-characteristic can be steady (at least in the subspace of the variables involved in the loop) for appropriate values of the parameters (Thomas, 1991 a and b). This was only a conjecture (based on the analysis of m a n y specific situations) until it recently became a theorem: see Section 3 of this work. 2.5. Logical identification of all steady states. How do we now identify the steady states of a system in practice? As regards the regular states, one just has to compare the state with its image: if they are identical, the state is steady. As Or, more generally, of the reunion of disjointed loops, I.e. of loops which share no variable.
IDENTIFICATION OF STEADY STATES
981
regards the singular states, in view of the theorem just mentioned, we consider only those which are loop-characteristic. As described in a more rigorous way (and with a more elaborate formalism) in Section 3, we reason that in the vicinity of a loop-characteristic state, the only operative interactions are those which constitute the loop itself. The characteristic state of an n-variable loop has 2" adjacent regular states, but it suffices to consider those two adjacent states whose images are maximal and minimal. To find these two states, one simply gives to each variable the values just above and just below the threshold considered if the variable acts positively, and the values just below and just above the threshold if the variable acts negatively.
x~y~
+1
z
For example, for the loop t [, the characteristic state is 201030 -3 (whose orders, p, are 2, 1 and 3, respectively) and the two adjacent states whose images are maximal and minimal are 212 and 103, respectively. Let AIB~C 1 and A2B2C 2 be the images of these states. In order to be consistent with the steady state equations, the orders of the relevant thresholds must be comprised in the interval of the images; more precisely, in the present case one must have A 2 < 2 0 < A 1 , B 2 < 10
y
X
Y
1
1
K1
L2
2
2
K12
L12
(extracted from Table 1). As one can see, state 2020 will be steady iffthe images of the adjacent tates encompass 2020. More specifically, one must have K 1 and L 2 < 20; K12 and L12 > 20. As our parameters can take only the values 0, 1 or 2 (like the corresponding variable), this means that the conditions for 2020 to be steady are: K 1 , Z 2 = 0 or 1 K12, L 1 2 = 2 .
(2) The characteristic state of loop +x~,) 1 is 1 0 - ; we thus have to consider the various possible values of y. Each of states 100, ~01 and ~02 will thus be
982
E.H. SNOUSSI AND R. THOMAS
considered for stationarity, and for each of them we will consider the two adjacent states and their images. For state ~00 the table is:
For state 101 the table is:
For state 102 the table is:
x
y
X
Y
x
y
X
Y
x
y
0 1
0 0
0 K1
0 0
0 1
l l
0 K1
L2 L2
0 1
This state is steady iff: K 1 = 1 or 2.
This state is steady iff: L 2= l and K~= 1 or 2.
X
Y
2
K2
L2
2
Klz
L2
This state is steady iff: L 2=2, K 2 = 0
and K12 = 1 or 2. N o t e t h a t t h e c o n s t r a i n t s a r e c o n t r a d i c t o r y for ~ 0 ~
(Klz--~-1) a n d 1 0 Q ~
( K 1 = 2); i n c o n t r a s t , e a c h o f t h e s e s t a t e s c a n c o e x i s t w i t h Q 0 0 ) . (3) F o r t h e c h a r a c t e r i s t i c s t a t e - 10, w e h a v e t o c o n s i d e r stateVs 010, 110 a n d 220: For state 010 the table is:
For state 110 the table is:
For state 210 the table is:
x
y
X
Y
x
y
X
Y
x
y
X
Y
0 0
0 1
0 0
0 L2
1 1
0 1
K1 K1
0 Lz
2 2
0 1
Kx Ka
L12
The condition of steadiness is: L 2 = 1 or 2.
The conditions are: L 2 = l or 2 and K1=1.
The steady states @
and @
Li
The conditions are: L t = 0 , L12=1 or 2, K1=2.
cannot coexist since @
requires
K 1 = 1 and (210) requires K 1= 2, but either of these states can coexist with
@
(4) As the loops +x ) l a n d
+y)1
are disjointed we have also to
consider their reunion, whose characteristic state is 1010. The conditions of stationarity, given by the table: x
y
X
Y
0
0
1
1
0 K1
L2
0
are simply K 1 and L 2 = 1 or 2. From this, one can see that for the set of parameters chosen above the system admits, in addition to the five stable states 00, 01, 10, 11 and 22 already mentioned, the six singular steady states 2020, 100, 101,010, 110 and 1010. A look at Fig. 2 shows that these logical steady states exactly fit with the steady states of the differential description.
I D E N T I F I C A T I O N O F STEADY STATES
983
Three remarks: (1) All this procedure is now completely informatized (Thieffry et al., 1992). Given the logical parameters a program returns to the logical steady states (and their nature). However, the most interesting approach consists of asking the parametric constraints for any characteristic state (or any combination of characteristic states) to be steady. (2) Given the logical parameters, one can easily find consistent real parameters and inject them into a set of differential equations (with step or sigmoid interactions). As described in Thomas and D'Ari (1990), the relation between the logical and real parameters is as follows: For the stepwise linear equation: 2 = k(x > toO)- k _ x, the logical constraint K > m simply means that k > m0. If one uses sigmoid interactions, the relation is k >amO, in which the value of a depends on the steepness of the sigmoid (a = 1 for a step function, a = 2 for a smooth sigmoid with a Hill number of 2). In fact, it is in this way that we have found the real parameter values to be used in Fig. 2. (3) Not only the location but also the nature of the steady state can be predicted on logical grounds. (4) For pedagogic reasons, we have reasoned here in terms of homogeneous logical equations. In practice, we always consider the possibility that a gene can have a residual activity even in less favourable conditions. Formally, this amounts to having an independent term in each logical equation: X=dx(klo + kllmx + ...). As already mentioned, all these operations can now be performed by appropriate programs (Thieffry et al., 1992).
3. Formal Description. 3.1. Notations and definitions. differential equations: dXi dt
We consider the system of piecewise linear
- kio+ ~ kijS~zJ(Xi/Oij)-kiXi, i=1
in which k i j ) O , ki>0, a o ~ { + ; - } S~'J(X/Oij)e{O, 1} is defined by:
i = 1 , 2 , . . . n;
(2)
defines the sign of the interaction and (chj--+
and
Xj>Oij
~ii---
and
Xj
=
984
E . H . SNOUSSI A N D R. THOMAS
The thresholds Ozj, corresponding to a given variable Xj, are different and are arranged in ascending order on the axis Xj, thus, to each threshold Ozj corresponds an order noted p~. The interaction graph which describes system (1) is an oriented, signed and weighted graph defined as follows. The n vertices are numbered from 1 to n and an arrow (j, i, cqj, Pij) expresses the fact that the ith equation depends on the variable Xj (i.e. kij ~ 0) acting above the threshold 0~ of the order p~ and eij is the sign of the interaction. The discrete m a p p i n g used in Section 2 to write the state formally is written as:
~o,(d(X))=di(Ooi(X)),
i= 1, 2 , . . . n,
(3)
with (I)i(X) = 1/k~[kio + ~ : l k~jS~'J(Xj/O~j)], i= 1, 2 , . . . n and d is the operator of space discretization defined by: d(X,,
X2 ....
X,) = (d,(Xx),
d2(X2),
. . .
d,(X,)),
with di(Xi)=r if and only if Xi is comprised between the two consecutive thresholds of respective order prz and p~+ 1,z. In a preceding paper (Snoussi, 1989) it was shown that the discrete m a p p i n g q) permits localizing all those stable nodes which are not located on a threshold. In the following we will show how to obtain all the steady states of system (1) using only the m a p p i n g q~. 3.2. Singular steady states. Definition 1. A state A is said to be singular if at least one of its components takes a threshold value. We note A =A(L J) in order to express that:
Viii, 3jeJ such that A i = Oji. A state not located on any threshold is called regular.
Definition 2. Let B = ( i 1 - - - ~ i 2 - - * • • • ir~il), a loop in the interaction graph. A singular state A is said to be characteristic of a loop B if: Vm~{1, 2 , . . . r}, Ai,=O,m+,,i m (m + 1 is the calculated m o d u l o r). In other words, A = A (/, J) is characteristic of a loop ifJis a permutation of/. More generally, a singular state is characteristic of a reunion of disjointed loops if it is characteristic of each loop. In Equations (2) and (3), the applications d, S + and S - had been defined only for the regular states. If one considers the singular states, it is natural to take:
IDENTIFICATION OF STEADY STATES
985
d,(Oj~)= Oj~, S~',(Ou/Oo)= ]0, 1 [ c and d,([a i, bi])= Ida(a,), d~(b~)].
Definition 3. A singular state A =A(L J) is said to be steady if for all XeN", such that d(X)=A, we have: ViCL A~= (Pi(A); VieI, ~,(X) is an interval ]a, b[; VieI, if Xj= Oij then Oijc]a, b[. This definition is justified by the fact that when Xj = O~j, S~',(Xj/Ou) is the interval ]0, 1[ and the differential equations (1) become equations of differential appartenance with steady state equations X~E @~(X) for a singular state. THEOREM 1. Any singular steady state is the characteristic state of a reunion of
disjointed loops. In other words, singular states which are not characteristic of a loop cannot be steady whatever the values of the parameters. Proof. Let A = A (/, J), a singular state which is not characteristic of a reunion of disjointed loops. By definition we have:
V,~L 3j~J: Ai=Oji; J is not a permutation of I. Then there exists, necessarily, an element ieI, which does not belong to J and thus:
3m6J such that A i = Omi with i~J. If X is such that d(X) = A, we have:
kio
1
Oi(X) = k i ~- ki
j:l
kuS~"(XjOij),
since i~J, then Xj• O,j for allje{1, 2 , . . . n} and Oi(X) is not an interval. This expresses that A is not a steady state. 3.3. Conditions ofstationarity of singular states. In this paragraph, we give a practical method to recognize whether a singular state characteristic of a reunion of loops is, or is not, steady. Without loss of generality, we consider the state: A = ( 0 2 1 , 0 3 2 , . . . Or,r_1, Ol,r, A r + l , . . .
A,),
986
E.H. SNOUSSI AND R. THOMAS
which is characteristic of the loop 1~2--, -.. --*r. A necessary condition for A to be steady is that:
n, Ai=qoi(A ).
for i = r + l , . . ,
Let pi be the order of 0~+ ~,~for i = 1, 2 , . . . r and X~N" such that d(X)=A. We have for i = 1, 2 , . . . r:
kio ki i- ~ S ..... 1 i ¢,(xt = ~ + ~ ~(xi_~/%_O+ ~ i=1 k,S~'J(X/%) jg:i-1
1 ~
kijSa'J(Xj/Oij )"
~- k / j = r + l
For j ¢ { 1 , 2 , . . . r } , Xj=Oj+I, j and when j # i - 1 , Xj#Oo, thus for j e {1, 2 , . . . r} a n d j # i - 1 we have S~'J(Xj/O,j) taking a value 0 or 1. F o r j e {r + 1 . . . . n}, Xj does not have a threshold value and S~'J(X/Oij) takes a value 0 or 1. Let:
K~ - k,o
1 ~
ki -t- ki
k,~S~,4X/O,) '
j=l
j¢i-1
we thus have, for ie{1, 2, . . . r}: ki i
• ~(X)=K~ + ~
1
S='-'.'(X,_l/Oi,i_l),
since X,_ 1 =Oi.i_l, S='-*.'(Xi_l/Oi,i_l)=]O, 1[ and: ¢~(X)=]Ki, G +
ki,i - 1 [
k~
"
The singular state A = d(X) is steady iff: for i~r+ l, q~i(A)=Ai; for i<~r, ~(X) is an interval which contains Oi+ 1,,. The first condition is given directly from the mapping q~ and for the second condition we must have:
K~
ki '
i.e.
IDENTIFICATION OF STEADY STATES
(4)
di(Ki)
987
}"
Let us calculate di(Ki) and di(K i + k~,i_ 1/ki) using only the regular states A - and A + defined as follows: A i- = A i for i e { r + l . . . . n} Ai- _ ~ Pi~ l - {,Pi
if o~i,i+ l = --{- for ie{1, 2 , . . . r}, if cq,i+ 1 = --
and Ai + = A i for i ¢ { r + 1. . . . n} Ai + =~pt t Pi-1
if if
-I- for ie{1, 2 , . . . r}. ai,i + l = --
O~i,i+ l =
Let X - and X + be such that d ( X - ) = A ie{1, 2 , . . . r}: Oi(X-
and d ( X + ) = A +, we have, for
) = Ki, i.e. di(Ki) = d,(@~(X-))
= ~o,(d(X
))
= q0~(A - ),
and ~ ( X +) = G + ~ki i 1 i.e. di (Ki + k',i--,'~ k i /I =di(O'(X+ )) = ~ ( d ( X +)) = ~Pi(A + ).
Condition (3) becomes: qoi(A - )< pi <<.cPi(A +). We thus have the following result: THEOREM 2. A singular state A = A(I, J) which is characteristic of a reunion of disjointed loops is a steady state if and only if: q)i(A~) =A}, ~= + O r - , j b r iq~I; goi(A- ) < pi <~qgi(A + ) for i6I. This theorem gives us a simple method to recognize if a singular state is steady or not using only the discrete mapping (? or equivalently the state table.
988
E.H. SNOUSSI AND R. THOMAS
Example.
Let the interaction graph be: +1
<_-.-
2
%/ /
- 1
/ - 3
-2
4<
3 +2
Let
d=021, 042, 1,014. +1
-3
-1
This singular state is characteristic of the loop 1 ~ 2 ~ 4 ~ 1 with I = {1, 2, 4} and J = {2, 4, 1}. We have A - =0311, A ÷ = 1210 and:
(klo k,2 k2o k3o k!o'~, e(A-)=dkk, + kx'ke'G'k4J klO
k12
~(A +)=d\ k, + V( +
k~ k2o k2~ Go k4o k42) k I ' k2 q- k 2 ' k 3 ' k 4 q- k 4/]"
A is steady iff:
dlt(kl° -~
kll
d(k2°~
)
)"
4. Summary and Conclusions. Typically, a model is built specifically in order to account for a certain n u m b e r of pre-existing experimental data. However, except for especially simple cases, a model implies a n u m b e r of additional predictions, most of which usually remain implicit, often even undetected. This is especially the case in the field of biological regulation in which the frequent presence of multiple intertwined feedback loops renders predictions about the dynamics far from obvious.
I D E N T I F I C A T I O N O F STEADY STATES
989
In the analysis of the dynamics of such complex systems, a crucial element is the identification of the number, nature and location of the steady states, and also of the domains in the parameter space in which the system behaves in qualitatively different ways. Our previous work had introduced the notion of logical parameters (Snoussi, 1989) and of singular logical states (Thomas and D'Ari, 1990; Thomas, 1991 a and b), which permit the identification of all the steady states on a logical basis. The concept of loop-specific state (Thomas, 1991 a and b) originates from the observation that when a feedback loop is effective (i.e. actually generates homeostasis or multistationarity), there is a steady state at the intersect of the thresholds of the loop. It was soon realized that among the singular states only those which are loop-specific can be steady, and this is formally demonstrated in Section 3 of this paper. From the practical viewpoints the identification of the steady states is greatly simplified, because instead of being obliged to scan all the (often very many) singular states for steadiness one has to consider only those (often very few) which are loop-specific. In practice, it is extremely easy to identify loops and their characteristic state, using a program if required by the number of variables (Thieffry et al., 1991 ), and the derivation of all steady states is simple. From a more conceptual viewpoint: (1) negative feedback loops can generate homeostasis, and the criterion for efficient homeostasis is the presence of a steady state--more precisely, a focus--located at thresholds of the variables involved in the loop; (2) positive feedback loops can generate multistationarity,t and the criterion is the appearance of a steady state--at the thresholds of the variables involved in the loop; this saddle point is usually on the border between two or more basins (characterized each by a steady state), dependng on the number of positive feedback loops. The recent progress considerably clarifies the situation, if only by introducing the generalization that the criterion for the effectiveness of a loop is that its characteristic state be steady. According to these views, in the absence of an effective positive loop a system can only have a single steady state. This means that if this system has several negative loops, not more than one of these loops (or more generally one reunion of disjointed loops) can be effective; which one depends on the parameter values. We expect, and systematically find indeed, that in the absence of a positive loop the conditions of effectiveness of negative loops are mutually exclusive (except if they share no variable). If there is one effective positive loop (i.e. there is one positive loop-specific t W h a t IS still lacking is a formal demonstration that a positive loop is a necessary condition for multistationarity and a negative loop a necessary condition for homeostasis.
990
E.H. SNOUSSI AND R. THOMAS
steady state) the system has three steady states. The loop-characteristic state, which is a saddle point, is on a separatrix which partly or completely separates the space of the variables of the loop into two basins, each characterized by an additional steady state. If in addition to the positive loop there are negative loops, these additional steady states (or one of them) may be homeostatic (e.g. a focus located on the thresholds of the negative loop). How can one have many steady states? The simple recipe is to have several positive feedback loops. If there are only effective disjointed positive feedback loops, m, such loops generate 3" steady states, 2m of which are stable (thus, seven genes subject to a positive autoregulation can account for 27 = 128 stable cell types). Additional interactions usually reduce the number of steady states (Thomas and Richelle, 1988), except if they introduce additional positive loops, as in our example (1).
APPENDIX In order to be able to feed singular states into our logical equations, we have to give a meaning to the binary variables ix, 2x . . . . when x takes a real value located on a threshold. It is natural to extend the definitions in the following way:
ix=
0,1[ if x=lO, 2x= if x> ~O
0,1[ i f x = 2 0 , if x> 20
where ]0, 1[ is the real interval. The logical equataons already used before including the singular states have the general form:
A singular state is said to be steady if for the variables x not on a threshold we have X=x, and for variables on a threshold value 0, the right member of equation (1) is an interval which encompasses the real value 0. Let us illustrate this definition with example (1):
=dx(klllX+k122y) ~cy-d~(k21 x+k22 Y) --
2
1
"
How can we calculate the image of a singular state--say xy = 101 with the parameter values chosen above? For x = 10 and y = l we have ~x=]0, t [ 2 x = 0 , l y = l and 2y=0. Introducing these values m the logical equations, we obtain:
Y=dr(k22)=L2=l, and since y = 1, we have Y= y.
klllX+k122y=]O, k i l l ,
IDENTIFICATION OF STEADY STATES
991
and since K 1= 1, 10
and
X - d~(]0, k11[) =]0, Kt[ Y = L 2.
Since for this state x = 10 and y = 1 the state is steady if 0 < ~0 < K 1 (or K 1~>1) and L 2 = 1. For state x y = 1020, we have: ~x=]0, l [ 2 x = 0 , ~y= 1, 2y=]0, 1[, thus Y=dy(k22), which is not an interval whatever the value of k22. Thus, 1020 is not steady for any values of the parameters, as expected for a singular state which is not loop-characteristic.
LITERATURE Glass, L. 1975. J. theor. Biol. 54, 85-107. Glass, L. and J. S. Pasternak. 1978. Bull. math. Biol. 40, 27-44. Kauffman, S. A. 1969. J. Theor. Biol. 22, 437-467. Rashevsky, N. 1948. Mathematical Biophysics. Chicago: University of Chicago Press. Ratner, V. A. and R. N. Tchuraev. 1978. Prog. theor. Biol. 5, 81 127. Snoussi, E. H. 1989. Dynamics Stability Sys. 4, 18%207. Snoussi, E. H., R. Thomas and R. D'Ari. 1990. In: Biological Feedback. R. Thomas and R. D'Ari (Editors), Ch. 7. Boca Raton, Florida: CRC Press. Sugita, M. 1961. J. theor. Biol. 1,415-430. Thieffry, D., M. Colet and R. Thomas. 1991. Proc. 8th International Conference on Mathematical and Computer Modelling. Oxford: Pergamon Press, in press. Thomas, R. 1973. J. theor. Biol. 42, 563-585. Thomas, R. 1991a. T. theor. Biol., in press. Thomas, R. 1991b. Lecture Notes Bmmath., in press. Thomas, R. and R. D'Ari. 1990. Biological Feedback. Boca Raton, Florida: CRC Press. Thomas, R. and J. Richelle. 1988. Discr. appl. Math. 9, 381-396. Van Ham, P. 1979. Lecture Notes Biomath. 29, 326 343.
R e c e i v e d 3 N o v e m b e r 1991 R e v i s e d 31 J u l y 1992