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VOLUME 78, NUMBER 6 PHYSICAL REVIEW LETTERS 10FEBRUARY 1997

Chemical Kinetics is Turing Universal

Marcelo O. Magnasco

Center for Studies in Physics and Biology, The Rockefeller University, 1230 York Avenue, New York, New York 10021

(Received 20 February 1996; revised manuscript received 15 August 1996)

We show that digital logic can be implemented in the chemical kinetics of homogeneous solutions:

We explicitly construct logic gates and show that arbitrarily large circuits can be made from them. This

proves that a subset of the constructions available to life has universal (Turing) computational power.

[S0031-9007(97)02332-6]

PACS numbers: 87.10.+e, 89.80.+h, 82.20.Mj

Interest in chemical computation has followed four dif-

ferent paths. It is one of the natural extensions of discus-

sions about information and thermodynamics, which go

back to Maxwell demon arguments and Szilard’s work

[1–5]. It is also a rather natural extension to the ap-

plication of dynamical systems theory to chemical reac-

tions [6–8], in particular logic networks stemming from

bistable reaction systems [9]. A lot of effort has been de-

voted to trying to devise nonstandard computational archi-

tectures, and chemical implementations provide a distinct

enough backdrop to silicon [10–12]. Finally, in recent

years biology has presented us with what looks to be ac-

tual chemical computers: the enzymatic cascades of cell

signaling [13–15].

One of the ﬁrst questions that can be asked in this

subject is whether universal (Turing) computation can

be achieved within some theoretical model of chemistry;

the most immediate one is standard chemical kinetics.

This question has been recently studied in some detail

[16–22], and even subject to experimental tests [23]. In

[18–20], Hjelmfelt et al. argued quite convincingly that

building blocks for universal computation indeed can be

constructed within ideal chemical kinetics, and that they

could be interconnected to achieve computation. How-

ever, many difﬁculties still lie in the way. An issue

not addressed by Hjelmfelt et al. is structural stability:

the tolerance of a system to changes in parameters and

functional structure. In particular, “gluing” together two

groups of chemical reactions will have appreciable effects

on the kinetics of both groups; the basic unit and the cou-

plings used in [18–20] require case-by-case adjustment of

individual parameters for proper functioning.

The purpose of this Letter is to provide a slightly

more formal proof that chemical kinetics can be used

to construct universal computers. I will concentrate on

the “next” level of difﬁculty, which is that of the global

behavior of a fully coupled system and its structural

stability. I will do it through the simplest approach: I will

show that classical digital electronics can be implemented

through chemical reactions. Since my key problem in

this scheme is showing global consistency, and the proof

requires arbitrarily large circuits, I will have to show that

the output of one gate can be plugged into the input of

others for arbitrarily many layers, without degrading the

logic, keeping at all times full coupling.

We will need a power supply. I will deﬁne mine to

consist of two chemical species called high and low;

their concentrations will be kept clamped strongly out

of equilibrium, so an external reservoir is assumed.

This approximates the power supply in cells, the two

compounds ATP and ADP; the cellular “power plants”

keep their concentration as constant as feasible, nearly

6 decades away from equilibrium. Thermodynamics

requires the logarithm of the equilibrium constants to

lie in the (left) span of the stoichiometry matrix; it is

important that all reactions we use satisfy this constraint,

so that there are no “hidden” power supplies.

The very ﬁrst thing we need to consider is the trivial

gate, the signal repeater, which copies input onto output.

Any problems we encounter with it will recur for any

other gate. Let’s say a chemical species ais the input and

bthe output. We will need bto exist in two chemically

distinct forms, band b[24]. If bis a compound of higher

energy than b, we can couple its production to the power

supply, as in b1high %b1low;in the absence of

other reactions, fbggoes to a small value determined by

the rate of spontaneous decay in b%b. This is then

a sort of “capacitor,” which we charge with the power

supply. If then the reaction b%bis catalyzed by a,

a1b%ab %ab %a1b,(1)

then a“shorts” the capacitor and discharges it, increasing

the concentration of b. Hence when fagis low, fbgis

low, and when fagis high, fbgbecomes high, and the

transitions have certain rise and decay times determined

by the precise rates we use.

In Fig. 1 we see the output of simulating a chain of

several such gates with a°! b°! c°! d.... The

gates are all identical; the only change between them is

the name of the compound. The wave forms are dying

as we go down this chain: The difference between the

“high” and the “low” levels is becoming smaller and

smaller. So this network is not a suitable signal repeater.

Figure 2 shows the output of a similar simulation using

the reactions

2a1b%a2b%a2b%2a1b(2)

1190 0031-9007y97y78(6)y1190(4)$10.00 © 1997 The American Physical Society

VOLUME 78, NUMBER 6 PHYSICAL REVIEW LETTERS 10FEBRUARY 1997

FIG. 1. A cascade of identical signal repeaters a°! b°!

c°! , using Eq. (1). The input to fagis a square wave.

Top (small) panels show each signal individually with varying

scales, bottom (large) panel shows all signals simultaneously on

the same scale. The amplitude of the signal gets reduced very

rapidly.

(i.e., double stoichiometry on the input). We can see that

the amplitude of the pulses gets stabilized; both high and

low now approach amply separate levels [25]. I will now

prove that higher stoichiometry is essential.

All concentrations become stationary after some tran-

sients. If we plot these steady levels as a function of the

inputs, we get the classical plots shown in Fig. 3. These

diagrams represent the concentration of bas a function

of a, but also of cas a function of b, and so on. If we

call xnthe nth compound in the chain, then the diagram

shows xn11as a function of xn;nhere labels position on

the chain. This is a recurrence relation, also called a map.

This type of map is usually studied in the theory of

dynamical systems, where it represents some dynamical

law, and nlabels time. A large part of dynamical

systems theory is devoted to the asymptotic states, i.e.,

what happens at arbitrarily long times. In our case this

translates to “arbitrarily deep into the circuit,” which

is what we want to study. Dynamical systems theory

tells us that the only asymptotic states of maps which

are monotonically increasing and bounded (our case)

are steady states. The steady states (also called ﬁxed

points) of a map occur when xn11xn, i.e., when the

curve intersects the diagonal line. They can be stable or

unstable; stable (unstable) means that if some xnis near

the ﬁxed point, then, for m.n, the xmare nearer to

(farther away from) the ﬁxed point; this happens when

FIG. 2. A cascade of signal repeaters with double stoichiom-

etry [Eq. (2)]. Same conventions as Fig. 1. The amplitude of

the signal converges to a steady value.

the curve is shallower (steeper) than the diagonal at the

intersection.

In the case of stoichiometry one sS1dthere are

at most two ﬁxed points, and only one can be stable

[26]. For S.1there can be three ﬁxed points, the

two outer ones being stable, the middle one unstable.

We can propagate logic arbitrarily deep into the chain

FIG. 3. The steady-state concentration of the outputs of two

signal repeaters, S1[Eq. (1)] and S2[Eq. (2)] as a

function of the steady-state level of the input a. The diagonal

line is fagas a function of itself; the intersections of the two

curves with this diagonal are the ﬁxed points.

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VOLUME 78, NUMBER 6 PHYSICAL REVIEW LETTERS 10FEBRUARY 1997

if and only if we have at least two distinct stable ﬁxed

points, with each one corresponding to a distinct logical

state. But two stable ﬁxed points are possible only

for S.1.

Now the main conceptual problems have been solved.

The only remaining point is to construct explicitly a few

different gates (Fig. 4); if all of the gates are “built”

(i.e., the rates so chosen) so that their response is one

of our ﬁxed points when the inputs are at the ﬁxed points

then they will be globally compatible. Strictly speaking,

one needs only NAND, since all logical functions can be

constructed from it, but since each internal wire in the

circuit is a chemically distinct compound, it is desirable

to implement gates directly [27]. A precise deﬁnition of

the gates can be found elsewhere [28].

Adding is a problem that exempliﬁes rather nicely the

spirit of this work, because when we add, we have to

shift the “carry” digits to the next column. These can

accumulate to generate a cascade, so we need to be able

to propagate logic across an entire network. In order to

add two three-bit numbers (giving a four-bit number as

the output), we need to cascade three full adders. The

three-bit adder is shown in Fig. 5; it can add up to 717.

Ephemeral memory can be implemented rather directly,

but if the memory is supposed to be long-term, care

must be exercised. A ﬂip-ﬂop can be made by having a

compound in two states sc,cd, and then two inputs sa,bd

which catalyze conversion to the other state by coupling

to the power supply:

FIG. 4. The output of one implementation of the four classical

gates. aand bare the inputs. While there are artifacts, the

logic levels are still well separated. AND,OR, and NAND are

implemented directly, XOR is implemented as AND(NAND,OR).

a1c1high %··· %a1c1low (3)

and similarly for bsending c°! c. The lifetime of

this memory would appear to be the lifetime of the

uncatalyzed reaction c%c. However, such a mechanism

is not resistant to ﬂuctuations in the inputs; even a minute

amount of catalyst can reduce the lifetime dramatically.

In order to make memory stable, we need to make the

system prefer to be either all cor all c. There are many

ways to do this; for instance,

2c1c1high %··· %3c1low (4)

and vice versa [29]. The addition of these two self-

catalytic reactions makes the memory strongly robust (see

Fig. 6) and, in principle, inﬁnitely long lived even in the

presence of input ﬂuctuations; however, energy is drawn

from the power supply to “refresh” the ﬂip-ﬂop. There is

some resemblance to dynamic vs static RAM, and to the

self-phosphorylating enzyme CamK II [30], which might

be implicated in long-term memory in neurons.

I have shown one particular explicit implementation of

digital logic in chemical kinetics, and thus shown universal

computation capabilities. However, many questions still

remain open (which I will comment upon in some greater

detail elsewhere [31]): What is the interplay between in-

formation transfer and thermodynamics? Since no catalyst

FIG. 5. Numerical simulation of the three-bit adder: c

a1b. The lower traces are the three bits of input aand

the three bits of input b; the four upper traces are the four bits

of output c. The transients as the inputs are changed show the

delays in propagating carries. The ﬁve columns of different

inputs show: 0100,71714,2124,613

9, and 7118. The network has about 140 compounds in

290 reactions.

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VOLUME 78, NUMBER 6 PHYSICAL REVIEW LETTERS 10FEBRUARY 1997

FIG. 6. Two ﬂip-ﬂops; sta is a “static” ﬂip-ﬂop [Eq. (3)], and

dyn a “dynamic” one with autocatalytic stabilization [Eq. (4)].

Both can switch between states fast as the inputs aand bare

pulsed. At time 100 both inputs are set to 0.1; sta forgets its

state, while dyn does not.

is perfectly selective for its substrate, how robust are

computations under the massive cross talk of random

“unintended” reactions? Are there equivalents of the

gain bandwidth and other classic theorems of electronics?

And, presumably, many more.

I would like to thank A. Ajdari, G. Cecchi, D. Chate-

nay, J.-P. Eckmann, A. Libchaber, G. Stolovitzky, and

D. Thaler for many stimulating discussions. Completed

in part at U. of Buenos Aires and Asoc. de Fisica Ar-

gentina; I want to thank G. Mindlin, S. Ponce, and J.P.

Paz for their hospitality. Supported in part by the Mathers

Foundation.

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[24] Compounds which exist in two chemically distinct states

can be implemented as proteins which can be phosphory-

lated, or as compounds which can be localized in two

different places; for instance, Ca11 in the cytosol is

“chemically distinct” from Ca11 in the ER, with channels

and pumps playing the role of kinases and phosphatases.

[25] These repeaters show the equivalent of input and output

impedanceycapacitances. If fagis changed abruptly, it

bounces, due to capture and release from the bound

complexes ab and ab. Convergence fails on the last

steps of the cascade when the last element is not

“terminated.”

[26] For S1the curve is convex. A convex curve can be

intersected by a straight line at most twice; at most one

intersection can be stable.

[27] Actual kinases from enzymatic pathways can have more

than one phosphorylation site and do logic directly on the

protein; so biological cascades can be more compact than

the networks shown here.

[28] Online at http:yytlon.rockefeller.eduy

[29] This can be done with only one autocatalytic reaction, plus

a nonspeciﬁc decay like c°! c(a self-phosphorylating

kinase and a phosphatase); in the absence of other

interactions, either S.1or cooperativity is required.

See also [9].

[30] H. Schulman, Curr. Opin. Cell Biol. 5, 247– 253 (1993).

[31] A full version of this paper, including a careful description

of the open problems, will appear elsewhere.

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