Computing Algebraic Functions with Biochemical Reaction Networks

Computing Algebraic Functions with Biochemical Reaction Networks

H. J. Buisman** Technische Universiteit Eindhoven

H. M. M. ten Eikelder** Technische Universiteit Eindhoven

P. A. J. Hilbers** Technische Universiteit Eindhoven

A. M. L. Liekens*,** Abstract In biological organisms, networks of chemical reactions control the processing of information in a cell. A general approach to study the behavior of these networks is to analyze common modules. Instead of this analytical approach to study signaling networks, we construct functional motifs from the bottom up. We formulate conceptual networks of biochemical reactions that implement elementary algebraic operations over the domain and range of positive real numbers. We discuss how the steady state behavior relates to algebraic functions, and study the stability of the networks’ fixed points. The primitive networks are then combined in feed-forward networks, allowing us to compute a diverse range of algebraic functions, such as polynomials. With this systematic approach, we explore the range of mathematical functions that can be constructed with these networks.

Technische Universiteit Eindhoven

Keywords Artificial chemistries, algebraic functions, feed-forward networks

1 Introduction Many proteins in biological cells are responsible for the transduction of signals and the processing of a cell’s environmental information [9]. Phosphorylation cycles are elementary and common motifs in biological signaling networks. The cycle is composed of a phosphorylating kinase and dephosphorylating phosphatase, which respectively catalyze the activation and deactivation of other proteins. Cascading networks of such phosphorylation cycles offer a platform for cells to control complex cellular processes [24]. Understanding of the behavior of cell signaling networks has important pharmaceutical, therapeutical, and computational applications [18, 2]. As an example of a small network, chemotaxis in bacteria adopts phosphorylation cycles for the implementation of their chemical circuitry [8]. The network for chemotaxis in E. coli samples the cell’s environment for nutrients, and controls actuators to direct the cell to higher concentrations in nutritional gradients [20]. In higher organisms, networks may comprise 80 or more kinases and phosphatases [14]. With increasing complexity and the employment of feedback, the dynamics of such large networks often becomes nonlinear, and it is consequently hard to infer knowledge about their behavior. A promising approach to study the behavior of complex signaling networks is to recognize common modules whose behavior can be understood. Each of these motifs is then be replaced by supernodes that * Contact author. ** Department of Biomedical Engineering, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: [email protected]

n 2009 Massachusetts Institute of Technology

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act similarly, in a way that is mathematically comprehensive. Hierarchical integration can lead to understanding of complex networks [17, 23, 15, 16]. Numerous motifs that perform computational tasks have been discovered in biochemical reaction networks. They have been observed to compute Boolean operations and implement simple binary computers [1, 21]. Cell signaling networks are known to exhibit parallelism, the integration and amplification of signals, bistable behavior and hysteresis through feedback, and memory [6, 5, 7, 22]. In vitro molecular computations have been performed with gene expression networks [3]. Other studies employ in silico evolutionary algorithms for the discovery of conceptual networks that perform basic computations, or act as bistable switches, filters, and stable oscillators [10, 19]. Many engineering metaphors have been put forward as analogies to signaling networks, such as neural networks and analogue electronic circuits [7]. As an example, elementary operations such as addition, multiplication, integration, and amplification can be found as modules in the MAPK pathways [4]. Here, we continue upon this modular approach by constructing elementary motifs from the bottom up, rather than the classical method of dissecting real biochemical reaction networks from the top down. In accordance with methods from artificial chemistries [12, 11], we synthesize conceptual networks that carry out elementary mathematical operations, such as addition, subtraction, multiplication, division, and root extraction. The implementations of basic operators are then combined in feed-forward networks that can perform a diverse range of algebraic functions. This allows us to construct, for example, networks that compute polynomials. 2 Methods We consider networks of interacting chemical species as the framework for implementing algebraic computations. Catalysts participate in the network by activating and deactivating other species according to specific kinetic laws. Concentrations of designated catalysts serve as the inputs of our computations. If the concentrations of other species in the system reach an equilibrium, we consider these steady state concentrations as the output of the computation. In our chemical networks, a variable x denoting a positive real number is represented by the concentration of a chemical species X, by identifying x with [X ]. Given the kinetic laws of reactions that act upon these species, we can write the changes in concentrations as ordinary differential equations (ODEs). ODE representations of chemical reaction networks assume well-mixed reaction vessels with continuous state space and continuous time. In this text, we further assume that catalytic reactions are instant, according to mass-action kinetics, that is, no intermediate catalyst-substrate complexes are formed. The systems studied in this article rely on the assumption that resources (including non-activated catalysts and energy) required for reactions are abundant, or that they are replenished and kept constant by an external source. We can then incorporate the constant availability of resources and energy in the kinetic rates of the reactions. Similarly, we do not take waste species, such as decayed or deactivated catalysts, into account, as they do not interfere with the internal behavior of the network. In the discussion, we elaborate on the effects of our modeling assumptions on the behavior of the reaction networks. 2.1 Elementary Algebraic Operations 2.1.1 Identification As a first network, we consider a motif that allows us to copy an input concentration a to an output concentration x. This reaction network is depicted in Figure 1. Species A catalyzes the production of species X, with rate constant k1. The output species X decays with rate constant k2 to waste. Since we do not consider resource and waste species, we can write the reactions corresponding to this motif as k1

A

!

X

!:

2

A þ X;

k2

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Figure 1. Reaction network that outputs a concentration x of species X, which is linearly related to the concentration a of the input species A.

The reactions in this network do not obey mass conservation. We have assumed fixed resource concentrations to be part of the kinetic law and its rate. Waste is considered irrelevant for the internal dynamics of the system. We adopt ODEs to describe the change of concentrations a and x. Since we assume that no substrate-catalyst complex is formed during catalysis, the concentration of the input species A remains constant. Two reactions influence the concentration of X. New particles X are being produced at a rate that is proportional to the concentration a of input catalysts A and to the rate constant k1. The resource concentration is inherently part of k1. Secondly, species X decays, at a rate proportional to x and the reaction rate constant k2. The system of differential equations for the concentrations a and x in this basic reaction network is thus given by a˙

¼

0;

x˙

¼

k1 a  k2 x:

For initial concentrations a0 and x0, the solution for this set of differential equations at time t is given by a

¼

x

¼

a0 ; k1 a0  ðk1 a0  k2 x0 Þe k2 t : k2

This system converges asymptotically to the stable fixed point (ˆa, x) ˆ given by aˆ

¼

xˆ

¼

lim a

¼

lim x

¼

t!l

t!l

a0 ; k1 a0 ; k2

which is independent of the initial concentration of output species. According to the steady state behavior of the network, we say that the network computes x = (k1/k2)a in the long run. If the rate constants k1 and k2 are equal, the concentration of the output species X converges to that of the input species A. We say that this network identifies the value of x with that of a. This network serves as a basic element to implement the elementary operations. 2.1.2 Multiplication Consider the network in Figure 2(a), with input catalysts A and B and output particle X. The network is represented by reactions AþB

k1

!

A þ B þ X;

k2

X !: This system is a slight extension of the network representing the identification operation. For the production of output particles X, both catalysts A and B have to concurrently take part in the Artificial Life Volume 15, Number 1

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Figure 2 . Catalytic reaction networks for (a) multiplication and (b) division.

reaction. In similarity to the previous network, we can study the system’s behavior through a set of ODEs. The values of input concentrations a and b remain constant because of our assumption of instantaneous catalysis with mass-action kinetics. The differential equation for x is given by x˙

¼

k1 ab  k2 x;

whose solution is x

¼

k1 a0 b0  ðk1 a0 b0  k2 x0 Þe k2 t ; k2

with stable steady state xˆ

¼

lim x

t!l

¼

k1 a0 b0 : k2

If we set the rate constants k1 and k2 of this system to be equal, the resulting network computes the steady state output value x that is the product of the input values a and b. The construction of this network requires multiple distinct catalysts to be present simultaneously to catalyze the production of output species. In biology, such enzymatic reactions are however atypical. Proteins can form functional complexes to catalyze reactions, but the resulting behavior is not similar to a multiplication, as the input concentrations are altered as well. We show how to adapt the topology of the multiplication network, by splitting up the catalysis step into two consecutive identification operations, as in Figure 3. In this network, the input particle A catalyzes the production ˜ from a resource particle with rate constant k1. The second input catalyst B of an auxiliary particle X ˜ as a substrate to produce the output particle X with rate constant k2. takes this auxiliary particle X ˜ Both species X and X are subject to decay, according to rate constants k1 and k2. Under the assumption that k1  k2, the first identification network copies input a into x˜, at a fast rate. The ˜ to produce second part of the network, which is slower than the first, consumes auxiliary particles X ˜ output X. In this network, species X acts as a fast buffer to store the concentration of A, and the ˜ to provide the result of the computation. It is common for rate constants second part consumes X of biological enzymatic reactions to differ by many orders of magnitude, so the assumption k1  k2 is realistic. The result of the computation is a better approximation of x = ab as the ratio k1/k2 is larger. 2.1.3 Division Consider the reaction network in Figure 2b, which is again a slight adaptation of the identification network. The production of the output species X is catalyzed by the input A, and the output species is degraded by the catalyst B: k1

A ! A þ X; BþX 4

k2

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Figure 3. A network that employs a buffer to compute the product x of inputs a and b, with k1  k2.

For this system, the differential equation for x is given by x˙

¼

k1 a  k2 bx;

whose solution is x

¼

k1 a0  ðk1 a0  k2 b0 x0 Þe k2 b0 t ; k2 b0

with stable steady state xˆ

¼

k1 a0 : k2 b0

If k1 = k2, the steady state input and output values are related by x = a/b. In the case where b = 0, the system diverges with limt!l x = l, since the production of the particle X takes place indefinitely, without decay of X. 2.1.4 Addition In the network depicted in Figure 4(a), both inputs A and B independently catalyze the production of output particle X: A B X

k1;1

!

k1;2

!

A þ X; B þ X;

k2

!:

The differential equation for x is given by x˙

¼

k1;1 a þ k1;2 b  k2 x;

Figure 4. Catalytic reaction networks for addition (a) and subtraction (b).

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whose solution is x

¼

k1;1 a0 þ k1;2 b0  ðk1;1 a0 þ k1;2 b0  k2 x0 Þe k2 t ; k2

with steady state xˆ

¼

k1;1 k1;2 a0 þ b0 : k2 k2

If all rate constants are equal, the steady state input and output are related by x = a + b.

2.1.5 Subtraction Consider the network in Figure 4b. Input species A and B catalyze the production of an output particle X and auxiliary particle Y, respectively, both with rate constant k1. Species X decays proportionally to the rate constant k1, and particle Y does not decay. Species X and Y can react with each other proportionally to a rate constant k2, producing waste. The set of reactions for this network is k1

A

!

B

!

X

!;

X þY

!:

A þ X;

k1

B þ Y;

k1

k2

We can write the differential equations for values of x and y by x˙

¼

k1 a  k1 x  k2 xy;

y˙

¼

k1 b  k2 xy:

Note the nonlinearity of this ODE system. We have not found an explicit formula for its solution. We can, however, find the fixed points (x, ˆ yˆ) of the system where x˙ = y˙ = 0. In this case, we have a unique fixed point given by xˆ

¼

a0  b0 ;

yˆ

¼

k1 b0 : k2 ða0  b0 Þ

We can study the stability of the fixed point by analyzing the Jacobian of the system, evaluated at the fixed point. The Jacobian is given by 0 Jðx; yÞ

6

¼ @

k1  k2 y

k2 x

k2 y

k2 x

1 A;

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which, evaluated at the fixed point, equals 0 Jðx; ˆ yˆÞ

¼ @

j11

j12

j21

j22

1

0

A ¼ @

1 b0 k1  ak0 b 0

k2 ða0  b0 Þ

1 b0  ak0 b 0

k2 ða0  b0 Þ

1 A:

If the Routh-Hurwitz conditions j11 + j22 < 0 and j11 j12  j12 j21 > 0 hold, then all roots of the characteristic equation of J have negative real parts [13], which means that the fixed point is asymptotically stable. When a0 > b0, these conditions hold, and the system converges to the fixed point (x, ˆ yˆ), where x converges to a0  b0. When a0 = b0, the fixed point does not exist. When a0 < b0, the fixed point does exist, but it is unstable. Here, xˆ = a0  b0 is negative, representing a negative concentration of X. The fixed point cannot be reached, since (x, y) cannot leave the first quadrant, denoting states with positive concentrations. By considering the nullclines of the ODE, we can study the long term behavior of the system. For a0 V b0 and sufficiently large t, the solution must stay in the region k1a  k1x < k2xy < k1b. It is easily seen that if a0 V b0, we have x ! 0 and y ! l if t ! l. Consequently, the network computes

x

¼

8 b;

:

otherwise:

0

2.1.6 Square Root Consider the network in Figure 5a. In this network, the input species A catalyzes the production of the output species X. Two particles X react with each other and then produce waste. The reactions for this network are thus k1

A

!

X þX

!

k2

A þ X; :

The ODE for x is given by x˙

¼

k1 a  2k2 x 2

with solution x

¼

x0 þ

k1 a0  2k2 x02 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2k2 x0 þ 2k1 k2 a0 cothðt 2k1 k2 a0 Þ

Figure 5. Networks for square root (a) and general nth root (b).

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which converges to the stable steady state xˆ

¼

rffiffiffiffiffiffiffiffiffiffiffiffi k1 a0 : 2k2

With rate constants k1 = 2k2, the reaction network computes the square root of its input, with x ¼ The topology for this network has also been discovered with an evolutionary algorithm [10].

pffiffiffi a.

2.1.7 nth Root The network in Figure 5b represents a more general form of the root primitive, which enables us to compute nth roots with integer n > 0. The production of the output particle X is catalyzed by the input particle A. For the production of auxiliary particle Y, n  1 catalysts X must be present simultaneously. Species Y acts as catalyst for the decay of X. The set of reactions is given by A ðn  1ÞX

k1

! A þ X; k3

! ðn  1ÞX þ Y; k2

X þY

! Y; k3

! :

Y

Note that this network contains a negative feedback loop. The concentration y approximates the (n  1)th power of x, as it employs a multiplication network. Species Y feeds back into this system by decreasing the concentration of X. The system of ODEs for this network is x˙

¼

y˙

¼

k1 a  k2 xy;

 k3 x n1  y ;

with unique fixed point xˆ

¼

yˆ

¼

rffiffiffiffiffiffiffiffiffiffi n k1 a0 ; k2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n1 k1 n : a0 k2

The Jacobian of this system, evaluated at the fixed point, equals 0

qffiffiffiffiffiffiffiffiffi 1 k2 n kk12 a0 C B k2 B C B C; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  n2 @ A n k1 k3 k3 ðn  1Þ k2 a0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   n

Jðx; ˆ yˆÞ

¼

k1 k2 a 0

n1

for which the Routh-Hurwitz conditions hold, indicating that the eigenvalues of the Jacobian have negative real parts. Consequently, the fixed point p is ffiffiffiasymptotically stable. For equal rate constants k1 = k2, we say that the network computes x ¼ n a. 8

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2.2 Combining Primitives In the previous sections we have constructed all elementary algebraic primitives as reaction networks. By identifying the output of one network with the catalytic input of another network, we can couple these primitives. This allows us to construct feed-forward networks that compute more complicated algebraic functions. As a result of the coupling, the input of the second network is no longer constant, in contrast with the primitive networks. We now study whether the fixed points of combined networks correspond with the composed functions, and whether these steady states are also stable equilibrium states. Consider two reaction networks, a first network with input species A1, . . . , Ak and output species X1, . . . , Xl and second network with inputs B1, . . . , Bm and outputs Y1, . . . , Yn. As with the primitive networks, we assume that the input species catalyze reactions in the network according to mass action, and consequently do not change over time because of the network’s dynamics. Let the ODEs that describe the two networks be given by ˙x

¼

fða; xÞ;

y˙

¼

gðb; yÞ;

which converge exponentially to their unique equilibrium points xˆ ðaÞ and yˆ ðbÞ for given input values a and b, as with our primitive networks: The eigenvalues of the Jacobians have negative real parts. Suppose we construct a new network by identifying an output of the first network with an input of the second network. Without loss of generality we can assume that the output Xl of the first network is identified with the input B1 of the second network. Hence the combined network has ODEs ˙x

¼

fða; xÞ;

ð1Þ

y˙

¼

gððxl ; b2 ; :::; bm Þ; yÞ:

ð2Þ

Note that this network has a and b2, . . . , bm as inputs. We prove that the following composition theorem holds: For given inputs a and b2, . . . , bm, the combined network has a unique asymptotically stable fixed point (ˆxðaÞ, yˆ ðxˆ l ðaÞ, b2, . . . , bm)), and the convergence to this fixed point is again exponential. Because the second network does not change the concentration of xl, the first network is independent of the second. Consequently, for each vector a the point xˆ ðaÞ is the unique fixed point of Equation 1. Now the value of b1 = xˆ l ðaÞ is fixed. With this value yˆ ðxˆ l ðaÞ, b2, . . . , bm) is the unique fixed point of Equation 2, for given b2, . . . , bm. Next we consider the stability of steady state (ˆxðaÞ, yˆ ðxˆ l ðaÞ, b2, . . . , bm)). The Jacobian of the combined system is given by 0

J

Af1 Ax1

:::

B . B .. B B B Afl : : : B Ax1 B B ¼ B B B B B B B B 0 @

1

Af1 Axl

.. .

0

Afl Axl

Ag1 Axl

Ag1 Ay1

.. .

.. .

Agn Axl

Agn Ay1

:::

:::

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C C C C C C C C C: C C Ag1 C Ayn C C .. C . C A Agn Ayn

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In this Jacobian we can recognize the Jacobians of the two original systems, as well as additional terms of the form Bg*/Bxl. The Jacobian is of the form 0 J

¼

@

J1

0

1 A;

A J2 where J1 and J2 are the Jacobians of the original networks, A is an m l matrix, and 0 is an l m zero matrix. Due to this zero matrix, the eigenvalues of J are simply the eigenvalues of J1 and J2. Since both original systems show exponential convergence, all eigenvalues of J1 and J2 have negative real parts. Hence this also holds for J, which implies exponential convergence to the fixed point of the combined system. For multiple outputs of the first network that serve as inputs for the second network, or multiple networks that generate inputs for the second, the theorem and its proof can easily be extended. Recursively applying the theorem allows us to construct cascading networks, where the fixed points and convergence properties of the constituent networks are carried over to the more complex network. It should be pointed out that the use of a subtraction operator in a combined network can lead to an incorrect outcome for the whole network. Indeed, if the result of a subtraction is negative, the operator returns a 0 output. This result is carried over to consecutive computations. For a computation to be correct, all intermediate results must be positive. As an example, the reaction network version of x = (a  b) + b does not result in a if b > a, but generates the larger of a and b as output. 3 Results We present some functions that can be constructed out of elementary operations. For all of the following networks, we assume that the kinetic rate constants are all equal to 1, unless noted otherwise. The behavior of the networks remains the same if all rate constants are scaled by the same nonzero factor. 3.1 Single-Node Computations Multiple additions, multiplications, and divisions can be incorporated through multiple production and decay reactions for a single output species. Basic network elements with no intermediate output species can be constructed to compute functions of the form

x¼

abc þ

þ def þ

pqr þ . . . þ stu þ . . .

with input catalysts A, B, C, D, E, F, . . . , P, Q, R, S, T, U, . . . . As an example, the network depicted in Figure 6a computes x ¼ dabþc þef in the long run.

abþc Figure 6. (a) Network that computes x ¼ dþef with one output species and no auxiliary species. (b) Network that computes the rational root with y = x m/n.

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3.2 Rational Powers The computation of rational powers x = am/n with integers m and n is sometimes considered to be an elementary operation for constructing algebraic functions. However, it can be constructed from the pffiffiffiffiffi primitive operations as x ¼ am . The network in Figure 6b computes a rational power. The production of the output species X is simultaneously catalyzed by m particles A. The network is otherwise similar to the network to find nth roots.

3.3 Polynomials We now give a general scheme to construct polynomials of the form y = a0 + a1x + a2x 2 + : : : + an x n, for a given input concentration x of species X. For each integer power x i with 1 < i V n that is required for the computation of the polynomial, we introduce a species X i. For each of these species, we generate the network so that the steady state value of x i is indeed the ith power of x. The network given in Figure 7(a) generates these values with reactions X þX X2 X þ X2 X3

X þ X þ X 2;

! ! ;

X þ X 2 þ X 3;

! ! ;

X þ X n1

!

X þ X n1 þ X n;

Xn

!

:

We introduce an output particle Y to represent the value of the polynomial. This output particle decays according to the reaction Y

!

:

For each positive aj , with j > 0, add a reaction that adds the value of aj x j to that of y. In this reaction, X j acts as catalyst and forms the output species Y according to rate aj : aj

X j ! X j þ Y;

Figure 7. Elements to construct networks of catalytic reactions to compute polynomials (a – c). Example network (d) computes y = 1  x + 4x2  2x3.

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as depicted in Figure 7b. Terms of the polynomial with negative values ak (k > 0) have to be subtracted from the polynomial. This can be achieved by implementing the subtraction operator. For this purpose, we introduce an auxiliary species Z that does not decay, but reacts with Y to produce waste: Y þZ

!

:

As in Figure 7c, we add a reaction for each negative term of the polynomial with Xk

ak

!

X k þ Z:

Finally, we have to add the scalar a0 to the output, which can be achieved by the non-catalyzed reaction a0

! Y if a0 z 0, and a0

! Z for negative values of a0. The resulting network computes the correct value of the polynomial in the long run, if the resulting output is positive. For negative values, the output y of the network converges to 0, as was the case with the primitive operator for subtraction. Polynomials in multiple input variables can be constructed in a similar manner. The first step is to produce the monomials with nonzero coefficients. Based on the sign of these coefficients, the monomials are added to the output particle Y or the auxiliary particle Z, as before. As an example of an implementation of a polynomial, the network depicted in Figure 7c computes y for input concentrations of X, so that the relation y = 1  x + 4x2  2x3 holds in the long run. 3.4 Solutions of Quadratic Equations In the following example, we construct and study a network that is designed to compute the solutions of the quadratic equation ax2  bx + c = 0, with inputs a > 0, b z 0, and c z 0. Note, we require the linear term of the quadratic equation to have a negative coefficient, to guarantee roots with positive real parts. Figure 8 depicts the feed-forward network that implements the quadratic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi formula x ¼ ðb F b 2  4ac Þ=2a as a catalytic reaction network based on the primitives for algebraic operations. In this network, all of the algebraic primitives are connected in various ways. Two subtraction operators have been used, which may result in wrong answers if the intermediate result of a subtraction is negative. The first subtraction is used to compute b2  4ac. If 4ac > b2, the amount of catalyst for the square root equals 0, which is an error that is carried over by the consecutive computations. Consequently, the network outputs only the real parts of the roots of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the quadratic equation. The subtraction operator is also used in the last step for b  b 2  4ac, but pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the result of this subtraction is always positive, since b z b 2  4ac for positive inputs a, b, c. 4 Conclusion and Discussion In this article, we have shown that conceptual designs of catalytic reaction networks allow for the implementation of a diverse range of mathematical functions. As designated concentrations reach an equilibrium state, elementary algebraic operations describe their relation with concentrations of input catalysts. The reaction networks can be coupled in a feed-forward manner to facilitate more complex computations, in such a way that the composed function and stability properties are unaffected. 12

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Figure 8. The quadratic formula for finding (the positive real parts of ) the roots of ax2  bx + c = 0. Each of the species in the network has been given a name that represents its steady state concentration. The output species of the computation are highlighted with a black border.

The list of algebraic motifs that can be constructed with catalytic reaction networks is clearly more extensive than depicted here. By taking advantage of the specific behavior of the subtraction network, switches, Boolean functions, and step functions can be implemented. Even so, the list of mathematical functions that can be constructed with feed-forward networks appears to be limited. We have not found ways to construct networks of finite size that represent transcendental functions, such as exponential, logarithmic, or trigonometric functions. Indeed, approximations of these functions can be constructed with networks that represent series expansions, but for these to be exact, the networks have to be infinitely large. Except for the construction of nth root networks, we have mainly concentrated on the construction of feed-forward reaction networks that mimic algebraic computations. Further incorporation of feedback mechanisms can generate oscillatory behavior, but also allow for the construction of networks that compute the roots and inverses of functions expressed by other motifs. With the added facility of feedback, numerous applications from control theory can be implemented as catalytic reaction networks. Since the range of functions that can be constructed with catalytic reaction networks is very diverse, they provide a well-equipped platform for biological cells to perform all sorts of computations. Relating conceptual functional real cell signaling networks to motifs provides insights into the internal workings of biological cells. The proposed framework does not offer any directions for the actual implementation of these networks. That is hard, because of the high specificity of enzymes to various aspects of the chemical context of our reactions, in contrast with the parameter flexibility that is required by our approach. Note that we have not taken the consumption of energy of our networks into account. Neither have we explicitly studied the convergence speed of the networks, although the solutions of differential equations and the eigenvalues of the Jacobians imply exponential convergence near the fixed point. Both the energy consumption and the switching speed of the networks are important for the optimal functioning of real biological control systems. In future work, we want to study the in silico evolution of conceptual networks, which may lead to understanding of the evolvability of information-processing networks in biological cells, while optimizing energy consumption and switching rates. Due to the assumptions in our models of reaction networks, the steady state outcomes of the studied networks can easily be related to elementary algebraic functions. On dropping assumptions, the behavior of the network deviates from this idealized representation. If the assumption of massaction kinetics is dropped and reversible catalytic reactions are assumed to form substrate-catalyst Artificial Life Volume 15, Number 1

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complexes, as with Michaelis-Menten kinetics, both input and output concentrations are affected. As the stability of the complex decreases, the behavior of the network better approximates the exact algebraic relation of our current interpretation. We have also ignored concentrations of resource and waste species. The topologies of our elementary networks can be converted to phosphorylation cycles, where the combined resource and waste species represent the non-activated or non-phosphorylated substrates. In such a scheme, resources are replenished by decayed or deactivated catalysts, and the production of output catalysts is limited by the amount of available resources. As the system can become saturated with limited resources, the buffer motif acts as a switch rather than as a linear relation. In future work, we want to study the steady state behavior of such elementary phosphorylation networks, and how these correspond to their algebraic functions in our current framework. Although the behavior of biological networks deviates strongly from our current model, the algebraic interpretations of patterns (such as multiplication and addition) may carry over to more realistic implementations. Understanding of the basic motifs allows us to explore the space of functions that can be expressed with biochemical reaction networks, leading to better understanding of real biological signaling networks. Acknowledgments We would like to thank Marvin Steijaert, Dragan Bosˇnacˇki, Jeroen van den Brink, and Sjoerd Crijns for discussing and proofreading the contents of this article. This work is supported by the European Community through the Evolving Cell Signalling Networks in Silico (ESIGNET) project of the Sixth Framework Programme. References 1. Arkin, A., & Ross, J. (1994). Computational functions in biochemical reaction networks. Biophysical Journal, 67(2), 560 – 578. 2. Ball, P. (2000). Chemistry meets computing. Nature, 406, 118 – 120. 3. Benenson, Y., Gil, B., Ben-Dor, U., & Adar, R. (2004). An autonomous molecular computer for logical control of gene expression. Nature, 429, 423 – 429. 4. Bhalla, U. (2003). Understanding complex signaling networks through models and metaphors. Progress in Biophysics and Molecular Biology, 81(1), 45 – 65. 5. Bhalla, U., & Iyengar, R. (1999). Emergent properties of networks of biological signaling pathways. Science, 283, 381 – 387. 6. Bray, D. (1990). Intracellular signalling as a parallel distributed process. Journal of Theoretical Biology, 143(2), 215 – 231. 7. Bray, D. (1995). Protein molecules as computational elements in living cells. Nature, 376(6538), 307 – 312. 8. Bray, D. (2002). Bacterial chemotaxis and the question of gain. Proceedings of the National Academy of Sciences, 99(1), 123 – 127. 9. Cohen, P. (2000). The regulation of protein function by multisite phosphorylation, a 25 year update. Trends in Biochemical Sciences, 25(12), 596 – 601. 10. Deckard, A., & Sauro, H. (2004). Preliminary studies on the in silico evolution of biochemical networks. ChemBioChem, 5(10), 1423 – 1431. 11. Dittrich, P. (2001). On artificial chemistries. University of Dortmund. 12. Dittrich, P., Ziegler, J., & Banzhaf, W. (2001). Artificial chemistries, a review. Artificial Life, 7(3), 225 – 275. 13. Gantmacher, F. R. (1959). Applications of the theory of matrices. Wiley. 14. Gomperts, B. D., Kramer, I. M., & Tatham, P. E. R. (2002). Signal transduction. Academic Press. 15. Hartwell, L. H., Hopfield, J. J., Leibler, S., & Murray, A. W. (1999). From molecular to modular cell biology. Nature, 402(6761), C47 – C52. 14

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16. Kholodenko, B., Bruggeman, F., & Sauro, H. (2005). Mechanistic and modular approaches to modeling and inference of cellular regulatory networks. Topics in Current Genetics, 13, 143. 17. Kholodenko, B., Schuster, S., Rohwer, J., Cascante, M., & Westerhoff, H. (1995). Composite control of cell function: Metabolic pathways behaving as single control units. FEBS Letters, 368(1), 1 – 4. 18. Kitano, H. (2002). Systems biology, a brief overview. Science, 295(5560), 1662 – 1664. 19. Paladugu, S., Chickarmane, V., Deckard, A., Frumkin, J., McCormack, M., & Sauro, H. (2006). In silico evolution of functional modules in biochemical networks. IEE Proceedings Systems Biology, 153, 223. 20. Rao, C., & Arkin, A. (2001). Control motifs for intracellular regulatory networks. Annual Review of Biomedical Engineering, 3, 391 – 419. 21. Sauro, H., & Kholodenko, B. (2004). Quantitative analysis of signaling networks. Progress in Biophysics and Molecular Biology, 86, 5 – 43. 22. Tyson, J., Chen, K., & Novak, B. (2003). Sniffers, buzzers, toggles and blinkers: Dynamics of regulatory and signaling pathways in the cell. Current Opinion in Cell Biology, 15, 221 – 231. 23. Weng, G., Bhalla, U., & Iyengar, R. (1999). Complexity in biological signaling systems. Science, 284(5411), 92 – 96. 24. Wolf, D., & Arkin, A. (2003). Motifs, modules and games in bacteria. Current Opinion in Microbiology, 6(2), 125 – 134.

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