Short Note

1 downloads 0 Views 172KB Size Report
Department of Electrical and Computer Engineering .... gave a negative answer to this question. He showed .... The solution n4=1, n12=5 cited by Heidel et al. [1] .... Networks, Journal of King Abdulaziz University: Engineering Sciences, Under.
JKAU: Eng. Sci., Vol.19 No.2, pp: 41-49 (2008A.D. / 1429 A.H.)

[Short Note] On Limitations of Using Scalar Equations for Analyzing Synchronous Boolean Networks Ali M. Ali Rushdi and Sultan O. Saad Al-Otaibi Department of Electrical and Computer Engineering King Abdulaziz University, Jeddah, Saudi Arabia [email protected] and [email protected] Abstract. A total description of a synchronous Boolean network is typically achieved by a matrix recurrence equation. A simpler description is possible when such a matrix equation is replaced by a scalar equation or a reduced scalar equation. Unfortunately, the scalarequation technique suffers from several shortcomings and limitations, both in procedure and results. We discuss these shortcomings and support our arguments with two illustrative examples.

1. Introduction A Synchronous Boolean network is a set of n nodes, each of which is either in state 1 (on) or state 0 (off) at any given time t. Each node is updated at time t+1 by inputs from any fixed subset of the set of nodes according to any desired logical rule. Since the total number of possible network states is finite (2n) and the network changes states sequentially in discrete time steps, the network must necessarily return to a previously occupied state in a finite time (at most 2n time points). This means that all possible trajectories of the network consist of either cycles (loops) of any length from size one (a fixed point) to a maximum of 2n, or transient states leading eventually to a cycle. An ideal total description of the network (in which one accounts for all 2n states) can be realized only for small n, and would be unfeasible for most networks of interest that 41

42

Ali M. Ali Rushdi and Sultan O. Saad Al-Otaibi

usually have 100 or more nodes. Thus methods are desired that provide an adequate description of the network utilizing quantities of size n rather than size 2n. Heidel et al.[1] suggested that sometimes the matrix equations necessary to describe the logic of a given Boolean network can be reduced to a smaller set of higher-order scalar equations or even a single scalar equation, and that such a scalar equation is more transparent to analyze for cycles, than the original matrix equations. The term “scalar equation” is used to denote an ordinary recurrence equation for a particular node of a Boolean network. This means that a scalar equation is an equation that involves two or more time instances of a single scalar variable and possibly some Boolean operator(s), e.g., the equation x1t+3 = 1+ x1t. As a sequel of Heidel et al.[1], Farrow et al.[2] suggested that a linear reduced scalar equation be derived from the more rudimentary nonlinear scalar equation. The reduced scalar equation is typically a simpler but a higher-order equation. It is a two-term scalar equation that includes no Boolean operators and equates a latter instance of each scalar variable xi (1 ≤ i ≤ n) to an earlier instance of the same variable. The general form of the reduced scalar equation is xit+r = xit+s , 1 ≤ i ≤ n, r and s are the smallest integers such that r 〉 s.

(1)

Such an equation yields the immediate information that (a) the least common multiple of all cycle periods is (r-s), and (b) the longest possible transient trajectory of the network has length s. When we first came across the work of Heidel et al.[1] and Farrow et al. , we found it really appealing and we initially thought of it as seminal work that can lead to considerable improvement in the state of the art. Unfortunately, after deep study of this work we discovered that it has several not-so-obvious shortcomings and limitations that we discuss below. [2]

2. Limitation of Procedure A synchronous Boolean network of n nodes is completely described by a set of n coupled recurrence equations in the n scalar node variables, or equivalently by a matrix recurrence equation[3, 4]. Heidel et al.[1] proposed the use of an alternative decoupled- or scalar-equation

On Limitations of Using Scalar Equations

43

representation, but they did not present an algorithm or a general procedure for obtaining a scalar equation representation. They only presented ad hoc procedures especially tailored to their specific examples. The procedure outlined by Farrow et al.[2] for obtaining a reduced scalar equation from a nonlinear scalar equation is again an ad hoc one and is rather lengthy and tedious, which makes it really error prone. The procedure is illustrated by one major 11-variable example on cell growth taken from Huang and Ingber[5]. Despite the fact that the final result of this example is correct, we have found that most of the steps leading to it were erroneous. The errors in these steps are not likely to be simply typographical. In fact, we had to rewrite all steps from scratch to give a correct proof of the desired result (see pp: 116-130 of Al-Otaibi [6]). Farrow et al.[2] claimed to be working to create a computer algorithm to systematically find any and all possible types of scalar equations, nonlinear and reduced, of a Boolean network. However, they admitted their suspicion that their procedures might not be streamlined into an algorithm. In fact, they posed the question "If an algorithm can be found, then the next question will be: Can it be implemented for "large" networks and how large do we really mean?" In a comment on their paper, Zhao[7] gave a negative answer to this question. He showed that even the determination of the number of fixed points (cycles of length 1) for monotone Boolean networks and the determination of the existence of fixed points for general Boolean networks are both strong NP-complete, which means among other things that both problems are highly intractable, and that the best algorithms that can ever be devised for them are highly inefficient. 3. Limitation of Results Heidel et al.[1] gave many examples to demonstrate how a scalar equation, once obtained, can lead to useful information on the network cycles. We revisit two of their examples to show that the scalar equations are not sufficient by themselves to extract all the information they were supposed to produce. Example 1 The network in Fig. 1 is a simple case of a 6-node affine system (that has linear terms plus constant terms in the Reed-Muller expressions of its

44

Ali M. Ali Rushdi and Sultan O. Saad Al-Otaibi

next state functions). The cyclic structure of affine Boolean networks is completely understood in a general way[8, 9], but this network has been analyzed by Heidel et al.[1] via the scalar-equations technique. A

x1

x2

x3

x6

x5

x4

B x1 x2 0 0 1 1

x2 x3 0 0 1 1

x3 x4 0 0 1 1

x4 x5 0 0 1 1

x5 x6 0 0 1 1

x6 x1 1 0 0 1

ON

ON

ON

ON

ON

OFF

Fig. 1. The Boolean network of Example 1, and the truth of its excitations.

The logic for this network is expressed by the following equations: x1t+1 = 1 + x6t, x2t+1 = x1t, x3t+1 = x2t, x4t+1 = x3t, x5t+1 = x4t, x6t+1 = x5t

(2)

Heidel et al.[1] combined these equations into the scalar equation x1t+6 =1+ x1t, which corresponds to the reduced scalar equation x1t+12 = x1t. All network states lie on cycles of period 12 and there are no cycles of period six. Thus there are no cycles of period one (fixed points), two, or three since these would also have period six (1, 2, and 3 are divisors of 6). Thus all cycles have either period 4 or period 12. (4 is the only divisor of 12 that is not a divisor of 6). The total number of states is 26 = 64 = 4 n4 + 12 n12 + nt

(3)

where n4 is the number of period-four cycles, n12 is the number of periodtwelve cycles, and nt is the number of transient states. Unfortunately, Heidel et al.[1] jumped at this point to the conclusion that the network has n4 = 1, and n12 = 5. We argue that this conclusion is unwarranted, unless some independent evidence is forwarded to support it. Our argument goes as follows. Equation (3) indicates that nt is a multiple of 4, i.e., (nt/4) is an integer k such that 0 ≤ k ≤ 16. Hence equation (3) can be rewritten as

On Limitations of Using Scalar Equations

45

n4 + 3 n12 = 16 – k

(4)

For a specific value of k, (4) represents a straight line in the twodimensional space of n12 versus n4, which might suggest that (4) has an infinite number of solutions. However, both n4 and n12 are restricted to be non-negative integers, which means that (4) has a finite number of solutions. Figure 2 illustrates the straight line represented by (4) for a zero value of the integer parameter k. This line is drawn over a rectangular grid of n4 = integer and n12 = integer lines. Figure 2 shows that (4) has 6 solutions for k=0 (shown highlighted). We can obtain the set of all possible solutions by drawing a family of parallel straight lines representing (4) for various values of k. Since the network states cannot all be transient, the maximum value of k is actually 15 corresponding to a single period-four loop. Table 1 lists all possible solutions of (4), which turn out to be 56 solutions. The scalar-equation technique has no explicit way for distinguishing between these 56 candidate solutions.

Fig. 2. Representation of the equation n4 + 3 n12 = 16 – k, k = 0 as a straight line in the 2-dimensional space of n12 versus n4. Permissible nonnegative integer solutions of (n4, n12) are highlighted. Other solutions are obtained by moving the straight line downwards parallel to itself for other integer values of k (1 ≤ k ≤ 15).

The solution n4=1, n12=5 cited by Heidel et al.[1] has five distinct cycles of period 12 and one cycle of period four. This solution represents the actual network solution as can be verified by viewing the map of all possible trajectories for the network, as given in Heidel et al.[1]. Unfortunately, there is no evident way to single out this desirable solution out of the mathematically valid 56 solutions. The scalar-equation technique fails to identify the actual network solution without some

46

Ali M. Ali Rushdi and Sultan O. Saad Al-Otaibi

implicit help or pre-knowledge from an exhaustive exponential-cost technique. Even if one assumes that this particular network has no transient states (nt=0 and hence k=0), the scalar-equation technique does not produce a unique solution on its own but produces the 6 solutions highlighted in Fig. 2 and shown on the top row of Table 1. Table 1. A listing of all possible solutions of equation (4). k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

n4+3n12 = 16 – k 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Possible integer pairs (n4, n12) (16,0),(13,1),(10,2),(7,3),(4,4),(1,5) (15,0),(12,1),(9,2),(6,3),(3,4),(0,5) (14,0),(11,1),(8,2),(5,3),(2,4) (13,0),(10,1),(7,2),(4,3),(1,4) (12,0),(9,1),(6,2),(3,3),(0,4) (11,0),(8,1),(5,2),(2,3) (10,0),(7,1),(4,2),(1,3) (9,0),(6,1),(3,2),(0,3) (8,0),(5,1),(2,2) (7,0),(4,1),(1,2) (6,0),(3,1),(0,2) (5,0),(2,1) (4,0),(1,1) (3,0),(0,1) (2,0) (1,0)

No. of solutions 6 6 5 5 5 4 4 4 3 3 3 2 2 2 1 1

Example 2 A

x1

x2

x4

x3

B x1 x2 0 1 1 0

x2 x3 0 0 1 1

x3 x4 0 0 1 1

OFF

ON

ON

x2 0 0 1 1

x4 0 1 0 1

x1 0 0 0 1

AND

Fig. 4. The Boolean network of Example 2, and the truth table of its excitations.

On Limitations of Using Scalar Equations

47

The network shown in Fig. 4 is again taken from Heidel et al.[1]. It has the nonlinear equation description: x1t+1 = x2t x4t, x2t+1 = 1 + x1t, x3t+1 = x2t, x4t+1 = x3t

(5)

For this network, Heidel et al.[1] readily computed the scalar equation x1t+4=(1+ x1t+2)(1+ x1t). Thus the scalar equation separates into two parts where the state of x1 at t+4 depends on the state of x1 at both t and t+2. The state of x1 at even time points is dependent only on the previous two even time points, while the state of x1 at odd time points is dependent on the two previous odd time points. Thus the state of x1 at the first two even and the first two odd time points must be specified and then the system can move along the trajectory mandated by the logic. Regardless of how node x1 starts, it moves to the period 3 orbit (0,0,1) on the even time points, or on the odd time points. When both even and odd times are considered together, there are only two different orbits that are obtained for node x1; (0,0,1) or (0,0,0,0,1,1). That is, the value of node x1 is in either a period three or a period six orbit. Similarly, each of the nodes x2, x3, and x4 is in either a period three or a period six orbit. The total number of states is 24 = 16 = 3 n3 + 6 n6 + nt

(6)

where n3, n6, and nt are the numbers of period-three cycles, period-six cycles, and transient states. There are 8 possible sets for the possible values for these numbers, which are nonnegative integers, as shown in Table 2. Again in this example, the scalar-equation technique fails to identify the unique solution n3 = 1, n6 = 1, and nt = 7 that can be obtained by tracing the actual trajectories. Table 2. Possible values for the numbers of period-three cycles, period-six cycles and transient states. n3 n6 nt 5 0 1 3 1 1 1 2 1 4 0 4 2 1 4 0 2 4 3 0 7 1 1 7

48

Ali M. Ali Rushdi and Sultan O. Saad Al-Otaibi

4. Conclusions The scalar-equation technique for analyzing synchronous Boolean networks can be applied as a manual heuristic non-algorithmic procedure in dealing with small specific networks, wherein it can provide some transparency, insight and partial information about such networks. The technique does not suffice as a perfect replacement for the matrix technique, nor does it usually yield complete information about the dynamic behavior of a network of an arbitrary size. An open question that might warrant further exploration is whether the scalar-equation method can be improved by finding remedies for some of its shortcomings. References [1]

[2]

[3] [4]

[5]

[6]

[7]

[8] [9]

Heidel, J., Maloney, J., Farrow, C. and Rogers, J., Finding Cycles in Synchronous Boolean Networks with Applications to Biochemical Systems, Int. J. Bifurcat. Chaos., 13 (3): 535-552 (2003). Farrow, C., Heidel, J., Maloney, J. and Rogers, J., Scalar Equations for Synchronous Boolean Networks with Biological Applications, IEEE Transactions on Neural Networks., 15 (2): 348-354 (2004). Cull, P., Linear Analysis of Switching Nets, Kybernetik, 8 (1): 31-39 (1971). Rushdi, A. M. and Al-Otaibi, S. O., On the Linear Analysis of Synchronous Switching Networks, Journal of King Abdulaziz University: Engineering Sciences, Under Publication. Huang, S. and Ingber, I., Shape-Dependent Control of Cell Growth, Differentiation, and Apoptosis: Switching Between Attractors in Cell Regulatory Networks, Exp. Cell Res., 261: 91-103 (2000). Al-Otaibi, S. O., A Study of Synchronous Boolean Networks and Their Applications, Unpublished Master Thesis, ECE Department, King Abdulaziz University, Jeddah, Saudi Arabia (2006). Zhao, Q., A Remark on “Scalar Equations for Synchronous Boolean Networks with Biological Applications,” IEEE Transactions on Neural Networks., 16 (6): 1715-1716 (2005). Milligan, D. K. and Wilson, M. J. D., The Behavior of Affine Boolean Sequential Networks, Connect. Sci., 5 (2): 153-167 (1993). Wilson, M. J. D. and Milligan, D. K., Cyclic Behavior of Autonomous, Synchronous Boolean Networks: Some theorems and conjectures, Connect. Sci., 4 (2): 143-154 (1992).

49

On Limitations of Using Scalar Equations

                                       

               . !"    .         %            #$  %     &"% ' ( ) *  .$        +   , -%    +     + 34   5 $  + / 0 1-2   +      =  16 5- 7 89: , -% ;<  . 5 .>