REVERSE ENGINEERING AND AUTOMATIC SYNTHESIS OF METABOLIC PATHWAYS FROM
OBSERVED DATA USING GENETIC PROGRAMMING

Symposium ON COMPUTATIONAL DISCOVERY OF COMMUNICATABLE KNOWLEGDE

SUNDAY MARCh 25, 2001

CSLI Stanford

John R. Koza

Stanford Biomedical Informatics, Department of Medicine

Department of Electrical Engineering

Stanford University, Stanford, California

koza@stanford.edu

William Mydlowec

Genetic Programming Inc., Los Altos, California

bill@pharmix.com

Guido Lanza

Genetic Programming Inc., Los Altos, California

guido@pharmix.com

Jessen Yu

Genetic Programming Inc., Los Altos, California

jyu@cs.stanford.edu

Martin A. Keane

Econometrics Inc., Chicago, Illinois

<makeane@ix.netcom.com>

From CHAPTER 1 OF *Genetic Programming III: Darwinian Invention and
Problem SolviNG* (Koza, BENNETT, ANDRE, KEANE 1999)

\"Most techniques of artificial intelligence, machine learning, neural
networks, adaptive systems, reinforcement learning, or automated logic
employ specialized structures in lieu of ordinary computer programs.

\"These surrogate structures include if-then production rules, Horn
clauses, decision trees, Bayesian networks, propositional logic, formal
grammars, binary decision diagrams, frames, conceptual clusters, concept
sets, numerical weight vectors (for neural nets), vectors of numerical
coefficients for polynomials or other fixed expressions (for adaptive
systems), genetic classifier system rules, fixed tables of values (as in
reinforcement learning), or linear chromosome strings (as in the
conventional genetic algorithm).

From CHAPTER 1 OF *Genetic Programming III*  CONTINUED

\"Tellingly, except in unusual situations, the world\'s several million
computer programmers do not use any of these surrogate structures for
writing computer programs.

\"Instead, for five decades, human programmers have persisted in writing
computer programs that intermix a multiplicity of types of computations
(e.g., arithmetic and logical) operating on a multiplicity of types of
variables (e.g., integer, floating-point, and Boolean). Programmers have
persisted in using internal memory to store the results of intermediate
calculations in order to avoid repeating the calculation on each
occasion when the result is needed. They have persisted in using
iterations and recursions. They have similarly persisted for five
decades in organizing useful sequences of operations into reusable
groups (subroutines) so that they avoid reinventing the wheel on each
occasion when they need a particular sequence of operations. Moreover,
they have persisted in passing parameters to subroutines so that they
can reuse their subroutines with different instantiations of values.
And, they have persisted in organizing their subroutines into
hierarchies.

From CHAPTER 1 OF *Genetic Programming III*  CONTINUED

\"All of the above tools of ordinary computer programming have been in
use since the beginning of the era of electronic computers in the l940s.
Significantly, *none has fallen into disuse by human programmers*. Yet,
in spite of the manifest utility of these everyday tools of computer
programming, these tools are largely absent from existing techniques of
automated machine learning, neural networks, artificial intelligence,
adaptive systems, reinforcement learning, and automated logic.

\"On one of the relatively rare occasions when one or two of these
everyday tools of computer programming is available within the context
of one of these automated techniques, they are usually available only in
a hobbled and barely recognizable form.

\"In contrast, genetic programming draws on the full arsenal of tools
that human programmers have found useful for five decades. It conducts
its search for a solution to a problem overtly in the space of computer
programs.

\"Our view is that computer programs are the best representation of
computer programs. We believe that the search for a solution to the
challenge of getting computers to solve problems without explicitly
programming them should be conducted in the space of computer programs.

The *topology* of a network of chemical reactions

• the total number of reactions in the network,

• the number of substrate(s) consumed by each reaction,

• the number of product(s) produced by each reaction,

• the pathways supplying the substrate(s) (either from external sources
or other reactions in the network) to each reaction,

• the pathways dispersing each reaction\'s product(s) (either to other
reactions or external outputs), and

• an indication of which enzyme (if any) acts as a catalyst for a
particular reaction

The *sizing* for a network of chemical reactions

• all the numerical values associated with the network (e.g., the rates
of each reaction)

Our approach

• establishing a representation for chemical networks involving symbolic
expressions (S-expressions) and program trees that can be progressively
bred (and improved) by means of genetic programming,

• converting each individual program tree in the population into an
analog electrical circuit representing the network of chemical
reactions,

• obtaining the behavior of the individual network of chemical reactions
by simulating the corresponding electrical circuit,

• defining a fitness measure that measures how well the behavior of an
individual network matches the observed time-domain data concerning
concentrations of final product substance(s), and

• using the fitness measure to enable genetic programming to breed an
improved population of program trees.

Five different representations

• **Reaction Network: T**he blocks represent chemical reactions and the
directed lines represent flows of substances between reactions.

• **Program Tree:** A network of chemical reactions can also be
represented as a program tree whose internal points are functions and
external points are terminals. This representation enables genetic
programming to breed a population of programs in a search for a network
of chemical reactions whose time-domain behavior concerning
concentrations of final product substance(s) closely matches observed
data.

• **Symbolic Expression:** A network of chemical reactions can also be
represented as a symbolic expression (S-expression) in the style of the
LISP programming language. This representation is used internally by the
run of genetic programming.

• **System of Non-Linear Differential Equations:** A network of chemical
reactions can also be represented as a system of non-linear differential
equations.

• **Analog Electrical Circuit:** A network of chemical reactions can
also be represented as an analog electrical circuit. Representation of a
network of chemical reactions as a circuit facilitates simulation of the
network\'s time-domain behavior.

Illustrative Problem No. 1  phospholipid cycle

• 4 reactions that are part of the phospholipid cycle, as presented in
the E-CELL cell simulation model

• External inputs

• glycerol (C00116)

• fatty acid (C00162).

• cofactor ATP(C00002)

• Network\'s final product

• diacyl-glycerol (C00165)

• Catalysts

• Glycerol kinase (EC2.7.1.30),

• Glycerol-1-phosphatase (EC3.1.3.21),

• Acylglycerol lipase (EC3.1.1.23), and

• Triacylglycerol lipase (EC3.1.1.3)

• 2 intermediate substances

• sn-Glycerol-3-Phosphate (C00093)

• Monoacyl-glycerol (C01885)

Illustrative Problem No. 1  phospholipid cycle

INTERESTING TOPOLOGY

• 2 instances of a bifurcation point (where one substance is distributed
to two different reactions)

• External supply of fatty acid (C00162) is distributed

• External supply of glycerol (C00116) is distributed

• 1 instance of an accumulation point (where one substance is
accumulated from two sources)

• glycerol (C00116) is externally supplied and

• glycerol (C00116) is produced by the reaction catalyzed by
Glycerol-1-phosphatase (EC3.1.3.21)

• 1 internal feedback loop (in which a substance is both consumed and
produced)

• Glycerol (C00116) is consumed (in part) by the reaction catalyzed by
Glycerol kinase (EC2.7.1.30).

• This reaction, in turn, produces an intermediate substance,
sn-Glycerol-3-Phosphate (C00093).

• This intermediate substance is, in turn, consumed by the reaction
catalyzed by Glycerol-1-phosphatase (EC3.1.3.21).

• That reaction, in turn, produces glycerol (C00116).

Four reactions from the phospholipid cycle

![](media/image1.wmf){width="6.525in" height="3.3409722222222222in"}

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Illustrative Problem No. 2  synthesis and degradation of ketone bodies

• 3 reactions

• External inputs

• Acetoacetyl-CoA

• Acetyl-CoA

• Final product

• Acetoacetate.

• Catalysts

• 3-oxoacid CoA-transferase (EC 2.8.3.5)

• Hydroxymethylglutaryl-CoA synthase (EC 4.1.3.5)

• Hydroxymethylglutaryl-CoA lyase (EC 4.1.3.4)

• 1 intermediate substance

• INT-1

llustrative Problem No. 2  synthesis and degradation of ketone bodies

3 NOTEWORHTY TOPOLOGICAL FEATURES

• 1 instance of a **bifurcation point** (where one substance is
distributed to two different reactions)

• Acetoacetyl-CoA

• 2 **accumulation points**

• Acetyl-CoA is an externally supplied substance and is produced by the
reaction catalyzed by Hydroxymethylglutaryl-CoA lyase (EC 4.1.3.4)

• Acetoacetate is produced by the reaction catalyzed by 3-oxoacid
CoA-transferase (EC 2.8.3.5) and by the reaction catalyzed by
Hydroxymethylglutaryl-CoA lyase (EC 4.1.3.4)

• 1 **internal feedback loop** (in which a substance is both consumed
and produced) • Acetyl-CoA is consumed by the reaction catalyzed by
Hydroxymethylglutaryl-CoA synthase (EC 4.1.3.5).

• This reaction, in turn, produces an intermediate substance (INT-1)

• This intermediate substance is, in turn, consumed by the reaction
catalyzed by Hydroxymethylglutaryl-CoA lyase (EC 4.1.3.4).

• That reaction, in turn, produces Acetyl-CoA.

Three reactions involved in the synthesis and degradation of ketone
bodies

![](media/image2.wmf){width="4.847222222222222in" height="3.75in"}

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GENETIC PROGRAMMING

\(1\) Generate an initial population of compositions (typically random)
of the functions and terminals of the problem.

\(2\) Iteratively perform the following substeps (referred to herein as
a generation) on the population of programs until the termination
criterion has been satisfied:

\(A\) Execute each program in the population and assign it a fitness
value using the fitness measure.

\(B\) Create a new population of programs by applying the following
operations. The operations are applied to program(s) selected from the
population with a probability based on fitness (with reselection
allowed).

\(i\) Reproduction

\(ii\) Crossover (Sexual recombination)

\(iii\) Mutation

\(iv\) Architecture-altering operations

\(3\) Designate the individual program that is identified by result
designation (e.g., the best-so-far individual) as the result of the run
of genetic programming. This result may be a solution (or an approximate
solution) to the problem.

ARCHITECTURE-ALTERING OPERATIONS

• The individual programs that are evolved by genetic programming are
typically multi-branch programs consisting of one or more
result-producing branches and zero, one, or more automatically defined
functions (subroutines).

• The *architecture* of such a multi-branch program involves

• the total number of automatically defined functions,

• the number of arguments (if any) possessed by each automatically
defined function, and

• if there is more than one automatically defined function in a program,
the nature of the hierarchical references (including recursive
references), if any, allowed among the automatically defined functions.

• Architecture-altering operations enable genetic programming to
automatically determine

• the number of automatically defined functions,

• the number of arguments that each possesses, and

• the nature of the hierarchical references, if any, among such
automatically defined functions.

Automatic Synthesis of Analog Electrical Circuits

Lowpass Filter Circuit

Time domain behavior of a lowpass filter to a 1,000 Hz sinusoidal input
signal

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Time domain behavior of a lowpass filter to a 2,000 Hz sinusoidal input
signal

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Frequency domain behavior of a lowpass filter

![](media/image3.emf){width="4.613888888888889in"
height="3.3472222222222223in"}

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Lowpass filter created by genetic programming that infringes on GEORGE
Campbell\'s patent

![](media/image4.wmf){width="6.418055555555555in"
height="1.3458333333333334in"}

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Squaring Computational Circuit created by genetic programming

![](media/image5.wmf){width="6.45in" height="3.813888888888889in"}

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Rising ramp  1 of 4 time-domain signals used to create squaring
computational circuit

![](media/image6.wmf){width="6.250694444444444in"
height="2.3319444444444444in"}

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Output for rising ramp input for squaring circuit

![](media/image7.wmf){width="6.347916666666666in"
height="2.6909722222222223in"}

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Automatic Synthesis of Controllers

Evolved controller that infringes on Jones\' patent

![](media/image8.wmf){width="6.5152777777777775in"
height="2.1034722222222224in"}

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Automatic Synthesis of Antennas

Antenna design created by genetic programming

![](media/image9.wmf){width="6.516666666666667in" height="1.39375in"}

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One-substrate, one-product chemical reaction

• One chemical (the *substrate*) is transformed into another chemical
(the *product*) under control of a catalyst

![](media/image10.wmf){width="3.1284722222222223in" height="2.65625in"}

Changing concentrations of substances in an illustrative one-substrate,
one-product reaction

![](media/image11.wmf){width="5.825in" height="2.634027777777778in"}

Chemical reactions

• The action of an enzyme (catalyst) in a one-substrate chemical
reaction can be viewed as two-step process in which the enzyme E first
binds with the substrate S at a rate *k*~1~ to form ES. The formation of
the product P from ES then occurs at a rate *k*~2~. The reverse reaction
(for the binding of E with S) in which ES dissociates into E and S,
occurs at a rate of *k*~-1~.

$$E + S\begin{matrix}
\overrightarrow{k_{1}} \\
\overrightarrow{k_{- 1}}
\end{matrix}\text{ES}\overrightarrow{k_{2}}P + E$$

• The concentrations of substrates, products, intermediate substances,
and catalysts participating in reactions are modeled by various rate
laws, including

• first-order rate laws,

• second-order rate laws, power laws, and

• Michaelis-Menten equations

• Michaelis-Menten rate law for a one-substrate chemical reaction is

$\frac{d\lbrack P\rbrack}{\text{dt}} = \frac{k_{2}\lbrack E\rbrack_{0}\lbrack S\rbrack_{t}}{\lbrack S\rbrack_{t} + K_{m}}$.

$K_{m} = \frac{k_{- 1} + k_{2}}{k_{1}}$.

• Psuedo-first-order rate law

$\frac{d\lbrack P\rbrack}{\text{dt}} = \frac{k_{2}\lbrack E\rbrack_{0}\lbrack S\rbrack_{t}}{K_{m}} = k_{\text{new}}\lbrack E\rbrack_{0}\lbrack S\rbrack_{t}$,

$k_{\text{new}} = \frac{k_{2}}{K_{m}}$.

E lectrical circuit representing the illustrative
one-substrate-one-product enzymatic reaction

![](media/image12.wmf){width="7.851388888888889in"
height="3.827777777777778in"}

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Sum-integrator

![](media/image13.wmf){width="3.7840277777777778in" height="3.71875in"}

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Subcircuit for one-substrate Michaelis-Menten equation MICH_1

![](media/image14.wmf){width="2.986111111111111in"
height="2.6666666666666665in"}

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Subcircuit definition in SPICE for the one-substrate Michaelis-Menten
equation MICH_1

\*NETLIST FOR MICHAELIS-MENTEN MICH_1

XXM4 4 3 2 XDIVV

XXM3 6 5 3 XADDV

XXM2 7 8 4 XMULTV

XXM1 9 5 8 XMULTV

.SAVE V(2) V(3) V(4) V(5) V(6) V(7) V(8) V(9)

.END

One-Substrate, Two-Product Reaction

![](media/image15.wmf){width="3.8472222222222223in"
height="2.1944444444444446in"}

Circuit for illustrative one-substrate, two-product chemical reaction

![](media/image16.wmf){width="6.319444444444445in"
height="3.6805555555555554in"}

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Two-Substrate, One-Product Reaction

![](media/image19.wmf){width="3.875in" height="1.9583333333333333in"}

$$E + A + B = \begin{matrix}
\overrightarrow{k_{1}} \\
\overrightarrow{k_{- 1}}
\end{matrix}\text{ABE}\overrightarrow{k_{2}}P + E$$

• Michaelis-Menten rate law for a two-substrate chemical reaction is

$\text{Rate}_{t} = \frac{\lbrack E\rbrack_{0}}{\frac{1}{K_{0}} + \frac{1}{K_{A}\lbrack A\rbrack_{t}} + \frac{1}{K_{B}\lbrack B\rbrack_{t}} + \frac{1}{K_{\text{AB}}\lbrack A\rbrack_{t}\lbrack B\rbrack_{t}}}$.

• When *k*~-1~ \~ 0 and *k*~-1~ \<\< *k*~1~ \<\< *k*~2~, it is often
satisfactory to use a psuedo-second-order rate law such as

$$\text{Rate}_{t} = k_{1}\lbrack A\rbrack\lbrack B\rbrack\lbrack E\rbrack$$

Circuit for two-substrate, one-product chemical reaction

![](media/image20.wmf){width="4.597916666666666in"
height="2.286111111111111in"}

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Two-substrate Michaelis-Menten equation MICH_2

$\text{Rate}_{t} = \frac{\lbrack E\rbrack_{0}}{\frac{1}{K_{0}} + \frac{1}{K_{A}\lbrack A\rbrack_{t}} + \frac{1}{K_{B}\lbrack B\rbrack_{t}} + \frac{1}{K_{\text{AB}}\lbrack A\rbrack_{t}\lbrack B\rbrack_{t}}}$.

Subcircuit for two-substrate Michaelis-Menten equation MICH_2

![](media/image21.wmf){width="6.383333333333334in"
height="4.761111111111111in"}

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Two-Substrate, Two-Product Reaction

![](media/image22.wmf){width="3.111111111111111in" height="2.0in"}

Electrical circuit representing a two-substrate, two-product enzymatic
reaction

![](media/image23.wmf){width="4.388194444444444in" height="2.7875in"}

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Repertoire of Functions in Program Tree

Four chemical reaction functions

  --------------------- ----------------- -------------------- ------------
  Function              Substrates        Products             Arity

  CR_1_1                1                 1                    4

  CR_1_2                1                 2                    5

  CR_2_1                2                 1                    5

  CR_2_2                2                 2                    6
  --------------------- ----------------- -------------------- ------------

• Each chemical reaction function returns a list (of length 1 or 2)
composed of the reaction\'s one or two products.

• The one-argument FIRST-PRODUCT function returns the first of the one
or two products produced by the chemical reaction function designated by
its argument.

• The one-argument SECOND-PRODUCT function returns the second of the two
products (or, the first product, if the reaction produces only one
product).

Repertoire of Terminals in the Program Tree

• Substances

• externally supplied input substances

• intermediate substances created by reactions

• output substances

• Enzymes

• Numerical constants for the rate of the reactions

Program tree corresponding to metabolic pathway for phospholipid cycle

![](media/image24.wmf){width="6.495833333333334in"
height="3.9430555555555555in"}

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Representation OF phospholipid cycle as a Symbolic Expression

(PROGn

(CR_2_1

EC3_1_1_3

1.45

C00162

(FIRST-PRODUCT

(CR_2_1

EC3_1_1_23

1.95

C00162

C00116

C01885

)

)

C00165

)

(CR_2_2

EC2_7_1_30

1.69

C00002

(FIRST-PRODUCT

(CR_1_2

EC3_1_3_21

C00093

C00116

C00009

)

)

C00008

C00093

)

)

Representation OF phospholipid cycle as a System of Non-Linear
Differential Equations

• Supply of the network\'s final product, diacyl-glycerol (C00165)

$$\frac{d\lbrack C\text{00165}\rbrack}{\text{dt}} = 1\text{.}\text{45}\lbrack C\text{00162}\rbrack\lbrack C\text{01885}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}1\text{.}3\rbrack$$

• Supply and consumption of the intermediate substance Monoacyl-glycerol
(C01885)

$$\frac{d\lbrack C\text{01885}\rbrack}{\text{dt}} = 1\text{.}\text{95}\lbrack C\text{00162}\rbrack\lbrack C\text{00116}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}1\text{.}\text{23}\rbrack - 1\text{.}\text{45}\lbrack C\text{00162}\rbrack\lbrack C\text{01885}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}1\text{.}3\rbrack$$

• Supply and consumption of the intermediate substance
sn-Glycerol-3-Phosphate (C00093) in the internal feedback loop

$$\frac{d\lbrack C\text{00093}\rbrack}{\text{dt}} = 1\text{.}\text{69}\lbrack C\text{00116}\rbrack\lbrack C\text{00002}\rbrack\lbrack\text{EC 2}\text{.}7\text{.}1\text{.}\text{30}\rbrack\text{- }1\text{.}\text{19}\lbrack C\text{00093}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}3\text{.}\text{21}\rbrack$$

• Supply and consumption of cofactor ATP (C00002)

$$\frac{d\lbrack\text{ATP}\rbrack}{\text{dt}} = 1\text{.}5 - 1\text{.}\text{69}\lbrack C\text{00116}\rbrack\lbrack C\text{00002}\rbrack\lbrack\text{EC 2}\text{.}7\text{.}1\text{.}\text{30}\rbrack$$

• Supply and consumption of fatty acid (C00162)

$$\frac{d\lbrack C\text{00162}\rbrack}{\text{dt}} = 1\text{.}2 - 1\text{.}\text{95}\lbrack C\text{00162}\rbrack\lbrack C\text{00116}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}1\text{.}\text{23}\rbrack - 1\text{.}\text{45}\lbrack C\text{00162}\rbrack\lbrack C\text{01885}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}1\text{.}3\rbrack$$

• Supply, consumption, and production of glycerol (C00116)

$$\frac{d\lbrack C\text{00116}\rbrack}{\text{dt}} = 0\text{.}5 + 1\text{.}\text{19}\lbrack C\text{00093}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}3\text{.}\text{21}\rbrack - 1\text{.}\text{69}\lbrack C\text{00116}\rbrack\lbrack C\text{00002}\rbrack\lbrack\text{EC 2}\text{.}7\text{.}1\text{.}\text{30}\rbrack - 1\text{.}\text{95}\lbrack C\text{00162}\rbrack\lbrack C\text{00116}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}1\text{.}\text{23}\rbrack$$

Electrical circuit corresponding to the metabolic pathway FOR
phospholipid cycle

![](media/image25.wmf){width="6.498611111111111in" height="4.2875in"}

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Preparatory Steps For genetic programming

Program Architecture

• Each program tree in the initial random population (generation 0) has
one result-producing branch.

• In subsequent generations, the architecture-altering operations
(patterned after gene duplication and gene deletion in nature) may add
and delete result-producing branches to particular individual program
trees in the population.

• Each program tree may have four result-producing branches.

Function Set

F = {CR_1_1, CR_1_2, CR_2_1, CR_2_2, FIRST-PRODUCT, SECOND-PRODUCT}.

Terminal Set

T = {ℜ, INT_1, INT_2, INT_3, C00116, C00162, C00002, C00165, EC2_7_1_30,
EC3_1_3_21, EC3_1_1_23, EC3_1_1_3 }.

Fitness Measure

• Concentrations of each of the four enzymes (EC2.7.1.30, EC3.1.3.21,
EC3.1.1.23, and EC3.1.1.3) are varied in accordance with 6 different
time series patterns over 30 half-second time steps:

• Each individual chemical reaction network is exposed to 9 test cases.
Thus, there are 270 fitness cases (9 test cases, each consisting of 30
time steps).

• Each of the nine test cases is constructed by choosing four different
time series from the above set of six time series as the concentration
for the four enzymes (EC2.7.1.30, EC3.1.3.21, EC3.1.1.23, and EC3.1.1.3)

Variation in the levels of the four enzymes for the nine test cases.

  ---------- ---------------- ---------------- ---------------- ---------------
  **Test     **EC2.7.1.30**   **EC3.1.3.21**   **EC3.1.1.23**   **EC3.1.1.3**
  case**                                                        

  1          Slope-Up         Sawtooth         Step-Down        Step-Up

  2          Slope-Down       Step-Up          Sawtooth         Step-Down

  3          Step-Down        Slope-Up         Slope-Down       Step-Up

  4          Step-Up          Slope-Down       Step-Up          Step-Down

  5          Sawtooth         Step-Down        Slope-Up         Step-Up

  6          Sawtooth         Step-Down        Knock-Out        Slope-Up

  7          Sawtooth         Knock-Out        Slope-Up         Step-Down

  8          Knock-Out        Step-Down        Slope-Up         Sawtooth

  9          Step-Down        Slope-Up         Sawtooth         Knock-Out
  ---------- ---------------- ---------------- ---------------- ---------------

Fitness Measure  CONTINUED

• There is a total of 270 data points. The data was obtained from the
E-CELL cell simulation model

• The concentrations of all intermediate substances and the network\'s
final product are 0 at time step 0.

• Glycerol (C00116), Fatty acid (C00162), and ATP (C00002) are
externally supplied at a constant rate that is not subject to
evolutionary change during the run

Rates for three externally supplied substances

  -------------------------------------------------- --------------------
  **Substance**                                      **Rate**

  Glycerol (C00116)                                  0.5

  Fatty acid (C00162)                                1.2

  ATP (C00002)                                       1.5
  -------------------------------------------------- --------------------

• Fitness is the sum, over the 270 fitness cases, of the absolute value
of the difference between the concentration of the end product of the
individual reaction network (diacyl-glycerol C00165 for the first
network and Acetoacetate for the second network) and the observed
concentration (data). The smaller the fitness, the better.

• An individual that cannot be simulated by SPICE is assigned a high
penalty value of fitness (10^8^)

Control Parameters for the Run

The population size, *M*, is 100,000.

A maximum size of 500 points (for functions and terminals) was
established for each result-producing branch.

Results Phospholipid Cycle

• The fitness of the median individual from the population at generation
0 is 297.3. This individual scores 17 hits (out of 270)

Median individual of generation 0

![](media/image26.wmf){width="6.425694444444445in"
height="3.2736111111111112in"}

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Results Phospholipid Cycle

Best of generation 0

![](media/image27.wmf){width="6.425694444444445in"
height="3.939583333333333in"}

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Results Phospholipid Cycle

Best of generation 10

![](media/image28.wmf){width="6.485416666666667in"
height="4.319444444444445in"}

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Results Phospholipid Cycle

Best of generation 25

![](media/image29.wmf){width="6.485416666666667in"
height="3.3243055555555556in"}

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Results Phospholipid Cycle

Best of generation 120

![](media/image30.wmf){width="6.485416666666667in" height="3.3375in"}

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Results Phospholipid Cycle

• The best-of-run individual has fitness of almost zero (0.054). This
individual scores 270 hits (out of 270).

• Correct topology

• The rate constants of three of the four reactions of this network
match the correct rates (to three significant digits).

Network of chemical reactions for the best-of-run individual from
generation 225

![](media/image31.wmf){width="6.485416666666667in"
height="3.313888888888889in"}

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Results Phospholipid Cycle

Electrical circuit for the best-of-run individual from generation 225

![](media/image32.wmf){width="6.498611111111111in" height="4.2875in"}

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Results Phospholipid Cycle

• Rate of production of the network\'s final product, diacyl-glycerol
(C00165)

$\frac{d\lbrack C\text{00165}\rbrack}{\text{dt}} = 1\text{.}\text{45}\lbrack C\text{00162}\rbrack\lbrack\text{INT}_{2}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}1\text{.}3\rbrack$.

• Rate of production and consumption of the intermediate substance INT_2

$\frac{d\lbrack\text{INT}_{2}\rbrack}{\text{dt}} = 1\text{.}\text{95}\lbrack C\text{00162}\rbrack\lbrack C\text{00116}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}1\text{.}\text{23}\rbrack - 1\text{.}\text{45}\lbrack C\text{00162}\rbrack\lbrack\text{INT}_{2}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}1\text{.}3\rbrack$.

• Rate of production and consumption of the intermediate substance INT_1
in the internal feedback loop

$\frac{d\lbrack\text{INT}_{1}\rbrack}{\text{dt}} = 1\text{.}\text{69}\lbrack C\text{00116}\rbrack\lbrack C\text{00002}\rbrack\lbrack\text{EC 2}\text{.}7\text{.}1\text{.}\text{30}\rbrack\text{- }1\text{.}\text{17}\lbrack\text{INT}_{1}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}3\text{.}\text{21}\rbrack$.

• Rate of supply and consumption of ATP (C00002)

$$\frac{d\lbrack\text{ATP}\rbrack}{\text{dt}} = 1\text{.}5 - 1\text{.}\text{69}\lbrack C\text{00116}\rbrack\lbrack C\text{00002}\rbrack\lbrack\text{EC 2}\text{.}7\text{.}1\text{.}\text{30}\rbrack$$

• Rrate of supply and consumption of fatty acid (C00162) in the
best-of-run network

$\frac{d\lbrack C\text{00162}\rbrack}{\text{dt}} = 1\text{.}2 - 1\text{.}\text{95}\lbrack C\text{00162}\rbrack\lbrack C\text{00116}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}1\text{.}\text{23}\rbrack - 1\text{.}\text{45}\lbrack C\text{00162}\rbrack\lbrack\text{INT}_{2}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}1\text{.}3\rbrack$.

• Rate of supply, consumption, and production of glycerol (C00116) in
the best-of-run network

$$\frac{d\lbrack C\text{00116}\rbrack}{\text{dt}} = 0\text{.}5 + 1\text{.}\text{17}\lbrack\text{INT}_{1}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}3\text{.}\text{21}\rbrack - 1\text{.}\text{69}\lbrack C\text{00116}\rbrack\lbrack C\text{00002}\rbrack\lbrack\text{EC 2}\text{.}7\text{.}1\text{.}\text{30}\rbrack - 1\text{.}\text{95}\lbrack C\text{00162}\rbrack\lbrack C\text{00116}\rbrack\lbrack\text{EC 3}\text{.}1\text{.}1\text{.}\text{23}\rbrack$$

• Internal feedback loop in which C00116 is both consumed and produced

Results Phospholipid Cycle

In summary, driven only by the time-domain concentration values of the
final product C00165 (diacyl-glycerol), genetic programming created both
the topology and sizing for an entire metabolic pathway whose
time-domain behavior closely matches that of naturally occurring
pathway, including

• the total number of reactions in the network,

• the number of substrate(s) consumed by each reaction,

• the number of product(s) produced by each reaction,

• an indication of which enzyme (if any) acts as a catalyst for each
reaction,

• the pathways supplying the substrate(s) (either from external sources
or other reactions in the network) to each reaction,

• the pathways dispersing each reaction\'s product(s) (either to other
reactions or external outputs),

• the number of intermediate substances in the network,

• emergent topological features such as

• internal feedback loops,

• bifurcation points,

• accumulation points, and

• numerical rates (sizing) for all reactions.

• Genetic programming did this using only the 270 time-domain
concentration values of the final product C00165 (diacyl-glycerol,).

Results  Synthesis and Degradation of Ketone Bodies

One individual from generation 0 with several noteworthy topological
features

![](media/image33.wmf){width="4.569444444444445in"
height="3.5277777777777777in"}

BBB2542

Results  Synthesis and Degradation of Ketone Bodies

Best network of generation 0

![](media/image34.wmf){width="5.319444444444445in" height="3.75in"}

BBB2543

Results  Synthesis and Degradation of Ketone Bodies

Best network of generation 5

![](media/image35.wmf){width="4.847222222222222in" height="3.75in"}

BBB2544

Results  Synthesis and Degradation of Ketone Bodies

Best-of-run network of generation 97

![](media/image36.wmf){width="4.847222222222222in" height="3.75in"}

BBB2545

Future Work

• Improved Program Tree Representation

• Multiplication and Division Functions

• Null Enzyme

• Minimum Amount of Data Needed

• Opportunities to Use Knowledge

• Designing Alternative Metabolisms