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35

4.1Chemical Neurons and Logic Gates

A neuron is either on or off depending on the signals it has received. A chemical neuron [10–13] is a similar device. Consider the hypothetical reaction mechanism (fig. 4.1) with the following elementary reaction steps [1]:

I1i + Ci X1i + Ci
X1i + Bi X2i + Ai
X3i	+ Ai X4i + Bi
X3i	I2i	(4.1)

Starred species are held constant by buffering, or reservoirs, or flows. A biological example will be given shortly. The macroscopic rate equations are given by

J1i = k1Ci − k−1Ci X1i
J2i = k2X1i Bi − k−2Ai
J3i = k3X3i Ai − k−3Bi
J4i = k4 X3i − k−4
dX1i /dt = J1i − J2i
dX3i /dt = −J3i − J4i
dAi /dt = −dBi /dt = J2i − J3i	(4.2)

There is constant inflow of I1 and a constant outflow of X2i . The stationary states of this reaction system, obtained by setting the time derivatives of the intermediates equal to zero, are plotted in fig. 4.2. For small values of the catalyst concentration Ci the concentration of Ai is small and at Ci = 1 the value of Ai increases abruptly and remains large thereafter. (The numbers are arbitrary. We use roman letters for species and italic letters for concentrations.) The reaction mechanism represents a chemical neuron, either on or off, as some parameter changes.

Next we consider the construction of logic gates by coupling of chemical neurons [2]. For each neuron in the network there is one copy of the reaction mechanism (fig. 4.1); the neurons are mechanistically similar but chemically distinct. In fig. 4.3 we show a possible coupling of neuron j to neuron i. We choose an enzyme mechanism in which the concentration of either Aj or Bj becomes an activator or an inhibitor of an enzyme

Fig. 4.1 Schematic of reaction mechanism with elementary steps given in eq. (4.1) and mass action rate expressions in eq. (4.2).

36 DETERMINATION OF COMPLEX REACTION MECHANISMS

Fig. 4.2 Stationary states of reactants Ai and Bi as a function of the catalyst concentration Ci for the reaction mechanism in fig. 4.1. (From [1].)

Eij for Cij . Suppose that the state of the neuron j is to excite neuron i. We have
Eij + Aj = Cij									(4.3)
The total enzyme concentration E0 is
ij
Eij0 = Eij + Cij									(4.4)
With the equilibrium constant of the rapid binding of the enzyme and Aj ,
K =	Cij								(4.5)
	Eij Aj

Fig. 4.3 Schematic of two reaction mechanisms constituting neurons i and j , and the influence of neurons j , k, and l on i. All reactions are reversible, and the concentration of the species marked by * is held constant. The firing of neuron j inhibits the firing of neuron l, and neurons k and l (data not shown) also influence the state of neuron i. The firing of neuron i inhibits the firing on neuron k. (From [1].)

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37

we can solve for Cij :

	E0
Cij =		ij		(4.6)
	1
	1 +
		KAj

The contributions of all neurons acting on neuron i are additive and hence the total concentration of the enzyme Ci is

Ci = Cij (4.7)

j

For neurons j and k to perform an AND operation on neuron i we need an activation reaction, eq. (4.3), for each of the species Aj and Ak . The concentration of the catalyst Ci is then the sum of Cij and Cik with the result, for K = Eij0 = Eik0 = 1,

1

1

2/3

Aj

=

1, Ak

0

4/3

Aj

Ak = 1

Ci

=

+

=

=

=

(4.8)

1

1

0, Ak

1

1

=

=

+ 2A j

+ 2Ak

0

Aj = Ak = 0

We see that only for Aj = Ak = 1 does Ci exceed 1; that is, if and only if neurons j and k are excited, then and only then is neuron i excited. This is an AND gate. Other logic gates can be constructed similarly, such as OR, NOR, and so on, by choosing appropriate values of K and Eij0 in this simple model.

4.2Implementation of Computers by Macroscopic Chemical Kinetics

With macroscopic chemical kinetics we can construct a Turing machine, a universal computer, which can add two strings of 0 and 1, and hence can subtract, multiply and divide, and carry out complicated calculations, as done by hand computers, laptops, and mainframe machines (see [1–4]).

A different kind of computing device is a parallel machine that can also be implemented by means of macroscopic chemical kinetics. For this purpose we choose a bistable chemical reaction, the iodate–arsenous acid reaction:

2IO3− + 5H3AsO3 + 2H+ → I2 + 5H3AsO4 + H2O

(4.9)

When this reaction is run in an open system—a so-called continuous-flow stirred tank reactor, or CSTR (fig. 4.4), with continuous influx of reactants and outflow products and unreacted reactants—then for certain influx conditions the system may be in one of two stationary states far from equilibrium: one of high I2 concentration, made visibly blue by addition of some starch, and one of low I2 concentration, a colorless solution. Measurements of bistability and chemical hysteresis in this system are shown in fig. 4.5. Bistable reaction systems have some similarities with bistable electronic switches, as pointed out some years ago by Roessler (see cited references in [1–3]). With bistable electronic switches it is possible to build an electronic computer, and now

38 DETERMINATION OF COMPLEX REACTION MECHANISMS

Fig. 4.4 Schematic of a continuous-flow stirred tank reactor (CSTR). The reacting fluid in the beaker is stirred.

we show the construction of a parallel computer, a pattern recognition device, with bistable chemical systems.

Suppose we take 8 CSTRs, each run as shown in fig. 4.6 with the iodate–arsenous acid reaction, eq. (4.8). Each circle is a CSTR containing this bistable reaction. The arrows indicate tube connections among the 8 tanks through which the reaction fluid from one CSTR is pumped at a set rate into another CSTR. The widths of the lines are a qualitative measure of the rate of transport from one CSTR to another. Each isolated reactor can be in one of two stable stationary states; 8 reactors can be in 28 such states. By our choice of the pumping rates we determine how many stable stationary states there are in the coupled reactor system. The dark (white) circles denote a state of high (low) iodide concentration. The choices of pumping rates and stable stationary states

Fig. 4.5 Plot of measured (I−) versus inflow rate coefficient k0 in the iodate–arsenous acid reaction run in an open, well-stirred system, a CSTR. The arrows indicate observed transitions from one branch of stable stationary states to the other stable branch, as the inflow rate coefficient is varied, and define the hysteresis loop. (Taken from [21] with permission.)

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Fig. 4.6 Schematic of 8 CSTRs, each filled with the iodate–arsonous acid reaction, connected by mass flow pumps (arrows); the varying thickness of the arrows denotes varying mass flow rates. Dark circle: the tank is in a high iodide stationary state; white circle: the tank is in a low iodide stationary state.

are made by a Hebbian rule familiar in neural network theory [1,2], which provides stronger connections between reactor tanks of the same color and weaker connections otherwise.

Experiments were made on this system [8] and some results are shown in fig. 4.7. The selected stable stationary states of the system are given in the last column at t = 120 min: the first reactor is blue (marked in black in fig. 4.7), the second colorless (marked in white), and so on. The system was started at time t = 0, not in one of its stationary states but in one of its stationary states with 2 errors. The errors are in reactors 1 and 2, and the initial state is not one of the stationary states of the coupled system. We are giving the system a pattern of blue and colorless states, with two errors, and we ask the system to correct these errors and identify the correct pattern as one of its stationary

Fig. 4.7 Experiments on pattern recognition. Three patterns are stored in the network. The system is started with one of these patterns but with errors in cells 1 and 2. The correct pattern is recovered after 60 min. (From [8].)