PRL 107, 048301 (2011)

week ending
22 JULY 2011

PHYSICAL REVIEW LETTERS

Weber’s Law for Biological Responses in Autocatalytic Networks of Chemical Reactions
Masayo Inoue1 and Kunihiko Kaneko2
1

2

Cybermedia Center, Osaka University, Toyonaka, Osaka 560-0043, Japan
Department of Basic Science, Graduate School of Arts and Sciences, University of Tokyo and ERATO JST, 3-8-1 Komaba,
Meguro-ku, Tokyo 153-8902, Japan(Received 28 March 2011; published 21 July 2011)
Biological responses often obey Weber’s law, according to which the magnitude of the response
depends only on the fold change in the external input. In this study, we demonstrate that a system
involving a simple autocatalytic reaction shows such a response when a chemical is slowly synthesized by
the reaction from a faster influx process. We also show that an autocatalytic reaction process occurring inseries or in parallel can obey Weber’s law with an oscillatory adaptive response. Considering thesimplicity and ubiquity of the autocatalytic process, our proposed mechanism is thought to be commonly
observed in biological reactions.
DOI: 10.1103/PhysRevLett.107.048301

PACS numbers: 82.39.k, 05.45.a, 87.10.e

Adaptation is ubiquitous in biological systems. In perfect adaptation, some state variables of a biological systemchange in response to a change in the external conditions
but slowly come back to their original values [1]. Such
adaptation is observed in signal transduction, sensory,neural systems, and so forth. Several theoretical studies
have investigated such adaptation by using a set of ordinary
differential equations, in which certain variables return totheir original values independently of the parameter values
representing the external condition [2–5].
Responses in a biological system have another ubiquitous characteristic: the degree of response is often proportional to the ratio of an external stimulus before and after it
is applied, rather than the absolute difference. In otherwords, the response detects the fold change, which iscommonly known as Weber’s law. This kind of response
was discovered in the field of psychology, wherein it wasobserved that our sensory response is proportional to thelogarithm of the magnitude of an external stimulus.
Recently, Weber’s law was also observed to be obeyedby cellular responses in microorganisms [6]. Shoval et al.
[7] further defined fold-change detection (FCD) in a sensestronger than Weber’s law: not only the magnitude of the
response peak but also the entire relaxation profile overtime depends only on the change ratio of the stimuli. The
relaxation profile over time is identical as long as the ratio
is constant, irrespective of the absolute magnitude of stimuli. Shoval et al. also developed a theoretical model forsuch FCD by using a feed-forward gene regulation network. However, ubiquity of this FCD is not yet confirmed,
in contrast to universality of Weber’s law.
An adaptive response usually has two components: rapid
response to a change and slow relaxation to the originalstate. Oosawa and Nakaoka confirmed the existence of
these distinct timescales in the adaptive response of
Paramecium to chemotaxis and also the relevance of their
existence to chemotaxis [8] (see also [5]). It is therefore
0031-9007=11=107(4)=048301(4)important to elucidate the relevance of such a timescaledifference to Weber’s law for adaptation.
Most biochemical reactions are catalytic, while replication of molecules as well as allosteric enzymatic reactions
often involve autocatalytic regulation [9–11]. In thepresent Letter, we show that a simple system with autocatalytic reactions exhibits an adaptive response according toWeber’s law if the autocatalytic process is slow. Because
the autocatalytic process often involves several steps andrequires a long time for completion, the ubiquity of adaptive responses obeying Weber’s law of a cell can be easily
understood from this simple system. We also demonstrate
Weber’s law in a system of chained or parallel-connectedautocatalytic reactions. In addition, we explore oscillatory
adaptation with Weber’s law and establish a condition
for it. Finally, we discuss the relevance of the results tobiological responses.
First, we study a simple reaction model of two chemicals
X0 and X1 with the following autocatalytic reaction, as
introduced in [5]: X0 þ X1 ! 2X1 ; the model also includes(a) the synthesis of X0 from an external resource chemical
S as S ! X0 and (b) the degradation of X0 and X1 . By
representing the concentrations of the two chemicals as x0
and x1 and suitably scaling the time and concentration
variables, we get the rate equation as
dx0 =dt ¼ S  x0 x1  x0 ; dx1 =dt ¼ ðx0 x1  x1 Þ=: (1)Among the two fixed-point solutions of Eq. (1), we focus
on the steady state x0 ¼ 1, x1 ¼ S  1 [12]. According to alinear stability analysis, this state is stable when S > 1,
i.e., as long as x1 > 0. It should be noted that x0 is independent of S. The chemical concentration responds to theconcentration of the external signal, S; when S increases(decreases), x0 increases (decreases) from its steady-statevalue but later returns to the original value. Thus, X0
always shows an adaptive response, whereas the steadystate concentration of X1 changes with S.

048301-1

Ó 2011 American Physical Society

PRL 107, 048301 (2011)

To examine Weber’s law, we study the response of x0
when S changes as S0 ! pS0 ðp > 0Þ, and calculate thedependence of the peak value of x0 during this change in S.
We assume   1, which is required to ensure a fast
response and slow adaptation. Under the adiabatic limit,
x1 changes more slowly than x0 does. During the fast
response of x0 to the change in S, x1 can be assumed toremain at the steady-state value under the condition of
S ¼ S0 . Then, the peak value of x0 is obtained fromðdx0 =dtÞx0 ¼xpeak ¼ 0 by fixing the value of x1 to S0  1.
0

¼ p. In a
A straightforward calculation gives us xpeak
0
peak
is estimated fromsimilar way, the peak time t
ðdx0 =dtÞ  S  S0 x0 and tpeak is given by 1=S0 [13].
Hence, x0 changes from the original value x0 ¼ 1 toxpeak
¼ p and returns to the original value. This amplitude
0of the response depends only on p, i.e., the ratio of the shift
in S, and is independent of the S0 value. Thus, our model
demonstrates Weber’s law under the adiabatic condition.
The time scale for adaptation is estimated from theJacobi matrix at the fixed point (x0 , x1 ). As its eigenvalue
with a smaller absolute value is given by   1= for large
S, the time scale for adaptation is given by  and isindependent of S. However, the peak time tpeak is givenby 1=S0 . Thus, the temporal profile of the response
depends not only on p but also on S0 . In this sense, ourconsidered model does not satisfy the condition for FCD
in [7].
We note that the adiabatic condition need not be rigid inorder for Weber’s law to be demonstrated reasonably
well. Figure 1 shows a plot of the peak-value ratioðxpeak1
 x0 Þ=ðxpeak2
 x0 Þ obtained by multiplying S p0012fold from S0 or S0 , respectively, as a function of . The
ratio is close to unity; it is independent of the initial S0 as
long as =tpeak  S0 * 100. In a previous study, we
determined that for efficient chemotaxis [5], the ratio of
the relaxation time to the response time to sense theenvironmental changes is on the order of 100, at several
microorganisms such as Paramecium [8] and E. coli [14].

1.5
1.4

∆2/∆1

week ending
22 JULY 2011

PHYSICAL REVIEW LETTERS

Thus, as estimated above, Weber’s law is approximately
satisfied for this time scale.
Adaptation obeying Weber’s law is not restricted to thistwo-component reaction system; in fact, it is generally
observed in a system with autocatalytic reactions with aslower timescale. We study several extensions of autocatalytic reaction networks showing adaptation with Weber’slaw. First, we consider an autocatalytic reaction occurringin series with N chemicals, as S ! X0 ! X1 ! . . . !
XN1 . The rate equations are given by
dx0 =dt ¼ ðS  1 x0 x1  x0 Þ=0 ;dxi =dt ¼ ði xi1 xi  iþ1 xi xiþ1  xi Þ=i ði  0;N  1Þ;dxN1 =dt ¼ ðN1 xN2 xN1  xN1 Þ=N1 :(2)Then, the steady-state concentrations are obtained as x0 ¼
S=ð1 x1 þ 1Þ, xi ¼ ðiþ2 xiþ2 þ 1Þ=iþ1 ði  0; N  2Þ,and xN2 ¼ 1=N1 . Accordingly, both the steady-state
concentrations xN2 and xN2m (m > 1) are independent of
the external-signal concentration S, and these results show
the adaptive response to the change in S. Here, we also notethat the adaptation alone of the chemical xN2 depends
only on the reaction process XN3 ! XN2 ! XN1 . Aslong as this chain reaction is autocatalytic and XN2 increases with S, xN2 undergoes perfect adaptation. We canmodify other reaction processes while retaining the adaptive response of XN2 .
First, we study the case of N ¼ 3 in detail. For simplicity, we use i ¼ 1. Suppose xi > 0; the steady state isgiven by x0 ¼ S=2, x1 ¼ 1, and x2 ¼ ðS  2Þ=2, which islinearly stable if S > 2 and x1 shows adaptive response to
the change in S. The relaxation process to the state, however, depends on the time scale, as is confirmed from theeigenvalues of the Jacobi matrix (Fig. 2). To achieve anormal adaptive response with monotonic relaxation,
0  1 , 2 is required (for large S, 80 < 2 and 0 =2 <
1 =S < 2 =32 [15]): otherwise, the system would show anadaptive response with damped oscillation.
An adaptive response with damped oscillation has often
been observed in experiments, e.g., in [16,17]. In contrast,
theoretical models for such oscillatory adaptation have

1.3

2.5

1.2

3

1

10

100

τ

1000 10000

x1(t)

x1(t)

1

2

1

1.5

0

1

1

FIG. 1 (color online). Plot of ratio of maximum response
amplitude 2 =1 (ordinate axis) versus (abscissa axis). i ¼
xpeaki x0 with the change Si0 ! pSi0 . 2 =1 ¼ 1 implies the0
independence of the response amplitude from the Si0 value, orWeber’s law. Data with three different (S10 , S20 , p) sets are shown:ðS10 ; S20 ; pÞ ¼ ð5; 50; 2Þ for þ, ðS10 ; S20 ; pÞ ¼ ð5; 500; 2Þ for , andðS10 ; S20 ; pÞ ¼ ð5; 50; 10Þ for .

2

2

1.1

τ0=0.01, τ2=10
0

20

40
time

60

80

0.5

5

τ0=2, τ2=1
0

20

40

60

80

time

FIG. 2 (color online). Behaviors of an adaptive variable (x1 ) in
the N ¼ 3 case. Responses corresponding to S ¼ 100 ! 200 at
t ¼ 0 and S ¼ 200 ! 400 at t ¼ 40 are shown, respectively.
(Left) 0 ¼ 0:01, 1 ¼ 1, 2 ¼ 10 and (Right) 0 ¼ 2, 1 ¼ 1,
2 ¼ 1, zoomed in the inset.

048301-2

PRL 107, 048301 (2011)

1 2 =20  S0 =12;

102

1.3

1.1

10-2

1

10-4
10-4 10-2

)

1.2

1

1

τ0 / τ1

102 104

FIG. 3 (color online). Phase diagram with 0 =1 (abscissaaxis) and 2 =1 (ordinate axis). Ratio of the maximum response x1 with S ¼ 200 !
amplitude, 2 =1 , is shown. 2 ¼ xpeak
1
400 and 1 with S ¼ 100 ! 200.

1
0.1
0.01
10-3

ln(x1

1.4

peak 2S0
d lnx1 S0 ðp  1Þt
fx ðtÞ  x2 ð0Þg: (3)ðS2Þ1 2
dt
0 1
0
104

(4)

which explains the boundary condition in the lower-rightarea. Note that the peak value itself increases as 2 =0
increases. Hence, to achieve an oscillatory response with alarge amplitude, the value on the left-hand side of Eq. (4)should preferably be large. Then, an oscillatory responsewith Weber’s law with a relatively large peak will beobtained at the border of the above condition.
For N > 3 as well, xN2 shows an adaptive responsewith both monotonic relaxation and damped oscillation.
For the former case, by carrying out similar analysis with
N ¼ 3 case, Weber’s law is shown to hold under thecondition of 0 ;    ; N3  N2  N1 . According
to Eq. (2), xN2m (m > 1) also shows an adaptive response,and Weber’s law for xN2m is satisfied when
0 ;    ; N2m1  N2m  N2mþ1 , i.e., if all processes of influx are fast and only the next runoff process

), (t

1

 x1 ¼ 2ðp  1Þ depends only on the
Accordingly, xpeak
1
ratio of change in S and not on the S0 value itself. Thus, themodel given by Eq. (2) with N ¼ 3 obeys Weber’s law
under the limits of 0 =1 ! 0 and 2 =1 ! 1.
To examine the validity of the above approximation, we
numerically studied the peak-amplitude ratio of x1 undervarying 2 and 0 and constant 1 . This condition corresponds to the upper-left area in Fig. 3, where Weber’s law
is confirmed and x1 shows a monotonic relaxation. In
Fig. 3, however, we find another regime that (approximately) satisfies Weber’s law in the lower-right triangle,
where x1 shows a damped oscillation and the above adiabatic conditions are not satisfied. Indeed, in this case,
Weber’s law is well obeyed for large S and 0 .
We now consider the response in the case that S changes
as S0 ! pS0 ðp > 0Þ with 0  1 , 2 and S0  1. We
solve Eq. (2) with the initial condition x0 ð0Þ ¼ S0 =2,x1 ð0Þ ¼ 1, and x2 ð0Þ ¼ ðS0  2Þ=2 to obtain the firstpeak in x1 at t ¼ tpeak , which is expected to be tpeak  1
because S is sufficiently large. From the assumptionof large S and 0  1 , x0 is solved approximately as0x0 ðtÞ  S20 þ ðp1ÞS
t for small t. Then, we get0

Temporarily neglecting the latter term for small t, we
get x1  exp½S0 ðp  1Þt2 =ð20 1 Þ. By substituting thisin dðlnx2 Þ=dt ¼ ðx1  1Þ=2 , we get x2 ¼ x2 ð0Þ exp½ðp 
1ÞS0 t3 =ð60 1 2 Þ. Now, with this increase in x2 , the second term in Eq. (3) is no longer negligible, as a result of
which the sign of Eq. (3) changes to negative and x1 stops
increasing. Thus, the time of occurrence of the first peak in
x1 is estimated from the time when the first term is equal to
the second one as ðtpeak Þ2  121 2 =S0 . By substituting
this value in the expression of x1 , the peak value of x1 isroughly estimated as xpeakexp½f6ðp1Þ2 g=0 . Hence,1
the maximum amplitude depends on the change ratio of S
and not on S0 . Next, we numerically verified the dependence of the peak amplitude on ’s from the original
equation [Eq. (2)]. As shown in Fig. 4, the above approximations give good estimates for the peak position andexplain the validity of Weber’s law.
Note that we need to satisfy tpeak  0 as 0 takes thelargest value among the three in this area. Then, we getpeakrarely been explored. Our proposed model provides asimple example of an oscillatory adaptive response.
In an autocatalytic reaction chain as well, Weber’s law
for responses is obeyed for only a certain range of parameters. Here, we study the conditions for Weber’s law in the
N ¼ 3 case as an example. To analyze this case, we again
study the system response to the change in S as S0 !
pS0 ðp > 0Þ. By setting 0  1  2 , we study the dynamics of x1 and its peak value (xpeak1 ) during an adaptive
response. Since the change in x0 is much faster than that inx1 , x0 is assumed to take the equilibrium value definedfrom the present x1 value, and hence, x0 is obtained from
dx0 =dt ¼ 0 as x0 ¼ pS0 =ð1 þ x1 Þ for given x1 . However,x2 still remains at the equilibrium value under the condition
S ¼ S0 , i.e., x2 ¼ ðS0  2Þ=2. Using these values, xpeak
1
that satisfies ðdx1 =dtÞx1 ¼xpeak ¼ 0 is given by xpeak¼ 2p1.
1

τ2 / τ1

week ending
22 JULY 2011

PHYSICAL REVIEW LETTERS

τ0=100τ0=1000

10-4
10-5

0.01

0.1

τ2

1

FIG. 4 (color online). Comparison between results of numerical simulation (points) and approximate calculation (lines).
peak Þ2 are shown as a function of  with  fixedlnðxpeak2
1
1 Þ and ðtpeak
to 1. lnðx1 Þ with 0 ¼ 100ðþÞ and 0 ¼ 1000ðÞ and ðtpeak Þ2with 0 ¼ 100ðÞ and 0 ¼ 1000ðhÞ are taken from the numerical simulation. The estimates of lnðxpeak
1 Þ with 0 ¼ 100 (dottedline) and 0 ¼ 1000 (broken line) and ðtpeak Þ2 (straight line) in
the text are also shown.

048301-3

PRL 107, 048301 (2011)

PHYSICAL REVIEW LETTERS

is slow. Weber’s law with damped oscillation is also satisfied for large S and 0 .
We also studied the case of catalytic reaction networks
having parallel paths. Two types of extensions are possible.
In one extension, many terminal chemicals exist instead ofa single x1 in Eq. (1), which are reached in parallel from x0(S ! X0 ! X11 , X12 , X13 ;    X1k ). It can be straightforwardly
shown that variable x0 shows an adaptive response, since
the mean field of all xj1 ’s acts as the variable of x1 in theoriginal two-variable case. Moreover, in this type of extension, each catalytic reaction need not be autocatalytic;
that is, the reaction from x0 to xj1 can be catalyzed by xm
1
with j  m. Adaptation obeying Weber’s law is possible in
the adiabatic limit. The other extension includes reaction
paths in two or more rows, each of which is a series
reaction, as in the case of variable N (S ! Y0 ! A1 !
   AN ! Y1 and S ! Y0 ! B1 !    BM ! Y1 ). Here,both an input variable (y0 ) and an output variable (y1 )can show adaptive responses only when there are an odd
number of elements in each row (N and M). In this case,adaptation obeying Weber’s law is again possible, but there
usually are more restrictions on the parameter values.
In this Letter, we have demonstrated that a simple autocatalytic reaction process in a system leads to its adaptationthat obeys Weber’s law. We first confirmed such an adaptation in an autocatalytic reaction of two variables.
Coupling of one more variable to the reaction led tooscillatory adaptation obeying Weber’s law. It would beinteresting to explore such an adaptive response experimentally. Indeed, such an oscillatory adaptation was observed in cAMP concentration in Dictyostelium cells [16].
Generally, cells undergo autocatalytic reactions to replicate themselves, where complex autocatalytic reactions
are often slower than simple catalytic ones. For example, ina sequential reaction process for the synthesis of a biopolymer that has catalytic activity for this chain reaction, moretime would generally be required for a longer polymer.
Also, the autocatalytic regulation exists in a system withallosteric enzyme reaction [11], and the (chain) autocatalytic reaction as in our model is known to exist in glycolytic reaction [11,18], as well as in the response to cAMP in
Dictyostelium [19]. The results of this study suggest that
adaptive response obeying Weber’s law is generally observed in such systems with slow autocatalytic reactions.
Adaptation or habituation refers to a general property ofa biological system to exhibit homeostasis, whereas
Weber’s law is applicable to a system exhibiting a sensory
response over a wide range of environmental conditions.
The variation in an external parameter S will often increase
with its value. Then, if the response were just proportional
to the difference in the external parameter before and after
the input, it would be too sensitive to large S and would not
be able to generate a reliable response under external noise.
Thus, a response to fold change with Weber’s law isapplicable to sensory systems in general.

week ending
22 JULY 2011

Here, Weber’s law was obeyed in an autocatalytic reaction system with time scale differences, without any special design or tuning of the parameters. Because of its
simplicity, the present mechanism is expected to have awide range of applications. It may also offer fresh perspectives on Weber’s law in general, including psychologicaland neural perspectives. Indeed, our proposed model canbe regarded simply as a system with self-positive feedback
and is not necessarily restricted to chemical reactions.
The authors would like to thank K. Kamino, S. Sawai,
and K. Fujimoto for useful discussions. M. I. was partially
supported by the JSPS.

[1] D. E. Koshland, Jr., A. Goldbeter, and J. B. Stock, Science
217, 220 (1982).
[2] S. Asakura and H. Honda, J. Mol. Biol. 176, 349 (1984).
[3] B. E. Knox et al., Proc. Natl. Acad. Sci. U.S.A. 83, 2345
(1986).
[4] N. Barkai, U. Alon, and S. Leibler, C. R. Acad. Sci., Ser.
IV Phys. Astrophys. 2, 871 (2001); A. Levchenko and
P. A. Iglesias, Biophys. J. 82, 50 (2002); R. Erban and
H. G. Othmer, SIAM J. Appl. Math. 65, 361 (2004); W.
Ma et al., Cell 138, 760 (2009).
[5] M. Inoue and K. Kaneko, Phys. Rev. E 74, 011903 (2006).
[6] Cohen-Saidon et al., Mol. Cell 36, 885 (2009);
L. Goentoro and M. W. Kirschner, Mol. Cell 36, 872(2009).
[7] O. Shoval et al., Proc. Natl. Acad. Sci. U.S.A. 107, 15 995(2010).
[8] F. Oosawa and Y. Nakaoka, J. Theor. Biol. 66, 747 (1977).
[9] M. Eigen and P. Schuster, The Hypercycle (Springer, New
York, 1979); S. A. Kauffman, The Origins of Order(Oxford Univ. Press, New York, 1993); S. Jain and S.
Krishna, Proc. Natl. Acad. Sci. U.S.A. 98, 543 (2001).
[10] K. Kaneko and T. Yomo, Physica (Amsterdam) 75D, 89(1994).
[11] A. Goldbeter, Biochemical Oscillations and Cellular
Rhythms (Cambridge University Press, Cambridge,
England, 1997).
[12] For the other state x0 ¼ S, x1 ¼ 0, which is unstable for
S > 1, there is no adaptive response since dx1 =dt ¼ 0.
[13] The direct solution of ðdx0 =dtÞ ¼ S  S0 x0 is monotonic,
but with inclusion of higher-order term in 1=ðS0 Þ, x0
shows a peak at tpeak .
[14] S. M. Block, J. E. Segall, and H. C. Berg, Cell 31, 215(1982).
[15] The exact conditions are pxﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ¼ ðS2 ﬃÞ=fðS  2Þ0 g > 88Þ xðx ﬃ 8Þg=64x < 1 =0 <and Sfx2 þ 20x  8  ðx p
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Sfx2 þ 20x  8 þ ðx  8Þ xðx  8Þg=64x.
[16] T. Gregor et al., Science 328, 1021 (2010).
[17] A. J. Ricci, Y. C. Wu, and R. Fettiplace, J. Neurosci. 18,8261 (1998).
[18] B. Hess and A. Boitewx, Annu. Rev. Biochem. 40,
237 (1971); E. E. Selkov, Eur. J. Biochem. 4, 79(1968).
[19] P. N. Devreotes, P. L. Derstine, and T. L. Steck, J. Cell
Biol. 80, 291 (1979).

048301-4