THE UN I V E R S I TY OF M I CH I GAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Computer and Communication Sciences Department Technical Report COMPUTER SIMULATION OF A LIVING CELL: MULTILEVEL CONTROL SYSTEMS Roger Weinberg Bernard P. Zeigler with assistance from: National Science Foundation Grant No. GJ-519 Washington, D. C. and Department of Health, Education, and Welfare National Institutes of Health Grant No. GM-12236 Bethesda, Maryland and Department of the Navy Office of Naval Research Contract No. N00014-67-A-0181-0011 Washington, D. C. and U. S. Army Research Office (Durham) Grant No. DA-31-124-ARO-D-483 Durham, North Carolina administered through: OFFICE OF RESEARCH ADMINISTRATION December 1969 ANN ARBOR Distribution of This Document is Unlimited

COMPUTER SIMULATION OF A LIVING CELL: Multilevel Control Systems. by Roger Weinberg and Bernard P. Zeigler SUMMARY A simple bacterial cell (Escherichia coli) has been modeled, and the input-output behavior of the model has been simulated as a program in FORTRAN IV for an IBM 360/67 digital computer. Automata theoretic analysis of the homomorphic model underlying the computer simulation enables us to investigate the information content and complexity of the simulation and of the measurement space representing our data base. The simulated cell is able to adjust its enzymes and DNA to grow in different chemical environments using allosteric modification of enzymes, and repression of RNA synthesis. It grows at realistic rates and achieves limited metabolic stability. INTRODUCTION Availability of experimental and theoretical analyses of the systems operative in living and reproducing organisms, as well as excellent presentations of powerful and convenient computer techniques make a computer simulation of a living cell a logical endeavor at this time. In particular much is known about lthe biochemical behavior of bacterial cells. [Biological laboratory studies have been done elucidating control of DNA replication in bacteria (Eberle and Lark, 1969; Clark, 1968). The 1968 Cold

2 Spring Harbor Symposium in Quantitative Biology was devoted to papers concerning replication of DNA in micro-organisms (edited by Frisch, 1969). Lark has reviewed initiation and control of DNA synthesis (1969), and the subject is covered in books by Mandelstam and McQuillen (1968), Hayes (1968), and Davis et. al. (4968). Control of Enzyme production by repression of messenger RNA production by the DNA is still being studied (Umbarger, 1969), and original hypotheses modified. Modification of enzymes already present in the cell, as a control mechanism, is reviewed by Umbarger (1969), and Batta (1960), and the relationship of this type of control to energy relationships in the cell has been reviewed by Atkinson (1966), discussed in Control of Energy Metabolism edited by Chance et. al. (1965), and investigated by Murray and Atkinson (1968). The importance of the ATP/ADP ratio in energy control relationships is becoming apparent in these works. These subjects, as well as convenient presentation of data on metabolic pathways, appear in books by Bernhard (1968), Westley (1969), Reiner (1968, 1969) and Mahler and Cordes (1966). Mathematical analyses of enzyme modification utilizing computer techniques have been published by Walter (1969a, 1969b), Cennamo (1969), Griffith (1968), Heinmets (1964), and Yeisley and Pollard (1964). Relationships have been drawn between regulatory mechanisms in microbial cells, and in higher cells by Mitchison (1969), Tsanev and Sendov (1969), Comings and Kakefuda (1968), Britten and Davidson (.1969), Gause (1966), and Heinmets (1966), making plausible the extension of computer simulation studie's of a bacterial cell to studies of cancer in higher organisms (Tsanev and Scndov, 1969). Weinberg and Berkus (1969), Weinberg (1969a, b, c) and Stahl (1967) have modeled living cells as computer

3 programs. Formal aspects of self-reproducing systems are described in Waddington (1969), Burks (1969), Codd (1968), Mesarovic (1968), and von Neumann and Burks (1966). Techniques for automata theoretic numerical analysis, and also computer simulation are well described in Gordon (1969), Mize and Cox (1968), IBM Corporation Scientific Subroutine Package (1969), Knuth (1969), Wendroff (1969), Ginzburg (1968), Kalman, et. al. (1969) and Ulam (1966). The connection between molecular controls and evolutionary mechanisms has been outlined for a computer simulation by Weinberg and Berkus (1969) taking advantage of basic genetic mechanisms (Strickberger, 1968, Kimura, 1964), and theoretical analyses connecting econometric studies (Gale, 1967) to a general theory of adaptive systems (Holland, 1969a, b).] The present computer simulation of a living cell is the first effort to compare the predictions of hypotheses concerning a complete, functional cell with detailed laboratory data. In our attempt to model a simple cell at the biochemical level we had to confront the complexity of the metabolic.pathways present in even the simplest of cells. More than 3,000 different kinds of molecules are present in a complex spatial and functional relationship. This complexity had to be drastically reduced to permit a simulation involving five hundred instructions in FORTRAN IV for a 360/67 digital computer. To construct our simulation we lumped molecules into pools, and considered 1) metabolic topology, 2) functional relationships among cell structures and chemicals, and 3) type of experimental data available in the literature. This process of lumping is further described and justified from a systems viewpoint in the appendix. Metabolic topology was considered, and an attempt was made to lump together as a single entity only those molecules which could be drawn adjacent to one

4 another on the metabolic map (Figures 1, 2). For example, in a pathway A + B + C, chemicals, A, B, C, might be partitioned into A, and BC, but would not be lumped into AC, and B. Functional relationships between groups of molecules were extremely important, and the molecules lumped together in any one model entity were, in some way, functionally a unit (Figure 3). Thus, all the molecules produced in the breakdown of sugar to CO2 and water to produce energy were lumped in this model since they could be considered functionally as molecules intermediate in a chemical pathway used to produce energy. Later models will employ more refined partitions, for example to capture the subtle and important relationships between molecules at different points in the glycolytic pathway citric acid cycle and cytochrome system. Experimental data were often available for large chemical pools, e.g. products of glycolysis and the citric acid cycle, making these separate entities logical candidates for grouping into single entities in this model of the cell. DESCRIPTION OF THE MODEL The simulation can be described by its state together with its transition function (Figures 1-5). The state of the cell is described 1) by the concentrations of thirty internal chemical pools, 2) by the genetic apparatus, and 3) by the cell volume. The transition function used to obtain the state of the cell at the next time step of the simulation from a given time step consists of difference equations and Boolean expressions describing 1) enzyme catalyzed chemical reactions, 2) allosteric modification of enzymes, 3) repression of

5 messenger RNA-production, 4) self-replication of DNA under genetic controls, and 5) permeability of the cell to the chemical pools represented in the simulation (Figure 3). A detailed description of an earlier version of the simulation is available (Weinberg and Berkus, 1969b). Input to the simulation consists of the concentrations of chemicals in the liquid environment in which the cell is growing. Output from the simulation consists of state descriptions during successive time increments. Comparison between simulation output and experimental data from the real world (Figures 5-15) enables us to judge the validity of the hypotheses used to write the simulation, to modify the hypotheses used to write the simulation in order to make the simulation more realistic, and to suggest critical real world experiments. The equations in the transition function are general, thus allowing us to simulate cell behavior in many different environments, and in changes from one environment to another. This is especially significant since we wish to test the stability of the model under a variety of conditions. RESULTS OF THE SIMULATION The environment simulated was liquid growth medium at a temperature of 37 degrees Centigrade, with an abundance of oxygen. The simulated experiments presented were in 1) minimal medium (medium containing glucose, ammonium salt, and minerals) and 2) broth (a rich medium containing additional amino acids, nucleosides and vitamins (Weinberg and Berkus, 1969b). The simulated cell grew faster in broth than in minimal medium, in agreement with laboratory data. This is a reasonable result, since addition of growth products implies that there are fewer molecules to be synthesized by the cell itself.

6 The simulated cell produced chemicals and cell mass at a logarithmic rate, but duplicated in a stepwise fashion (Figure 5) just as the real cell does. Since the simulated cell produced these smooth growth curves from a complex interaction of many equations, the growth curves are a good preliminary confirmation of the models used to write the simulation. The simulated cell employed repression to control the production of its enzymes (Figure 21, Appendix). Repression operated at the DNA level. For example EK2 was the enzyme pool needed for producing amino acids from carbohydrates. EK2 was produced under control of DNA by way of the RNA pools as long as the amino acid pool concentration was below a certain critical level. DNA directed the production of messenger RNA specific for the production of EK2. EK2 was produced by hooking together amino acids attached to transfer RNA. This hooking was done by messenger RNA attached to ribosomes. If the amino acid level rose above the critical level, production by DNA of messenger RNA responsible for EK2 production was sharply curtailed by the nature of the Boolean equations in the transition function. The messenger RNA already present rapidly decayed, and almost no new messenger RNA for EK2 production was formed. Of course if the amino acid concentration fell too low, insufficient amino acids were available for hooking together into the EK2 enzyme; production of all enzymes was blocked in the event of.extreme scarcity of amino acids as a result of the form of the differential equations concerning their production. The simulated cell employed feedback inhibition to control the activity of the enzymes already present (Weinberg and Berkus, 1969b). For example, EK2,

7 the enzyme for production of amino acids from carbohydrates, appeared in three different forms: pure enzyme, enzyme with one molecule of amino acid attached to it, and enzyme with two molecules of amino acid attached to it. These three forms of EK2 had different catalytic ability. The relative amount of EK2 in each form determined the activity of the EK2 present in the cell in terms of its efficiency in converting carbohydrate into amino acids. The percentage of EK2 in each of the three forms was determined by the number of amino acid molecules per cell volume unit (one cell volume unit was taken as the volume of a cell growing rapidly in mineral glucose medium with ammonium salt). The higher the amino acid concentration in the simulated cell, the greater was the percentage of EK2 in its low activity form, aid the less effective was the EK2 in production of amino acids from carbohydrates. We tested the ability of the simulated cell to grow in a medium it had never "seen" before by simulating a shift down to low glucose minimal medium -4 with 10 times the usual glucose concentration found in minimal medium (Figure 15). The simulated cell decreased its growth rate in this shift down, just as the real cell does (Moser, 1958). The cell could also adjust to shifts up from minimal medium to broth, and back down from broth to minimal medium when it was using its feedback controls. It is significant that without feedback controls the orderly shift up from minimal medium to broth was not possible (Figures 6-10). The simulation experiments to determine the function of the feedback controls was performed as follows: the simulated cell was shifted down from growth in broth to growth in minimal medium, and growth was followed for ten seconds. The results of the shift were plotted along with a broth control for

8 the simulated behavior obtained with and without feedback equations. Similarly, shifts up from minimal medium to broth were studied. Each graph represents measurements of some pool (such as ATP) during the shift, and during the corresponding control run. The simulated data agreed well with laboratory data when feedback controls were present (Figures 6-10) but the simulated cell without its feedback controls was no longer able to realistically handle shifts up from minimal medium to broth (Figures 11-14). This suggested that feedback in the real cell was evolved to handle shift up situations since the normal pathways are not stable in this condition without feedback. The simulated cell with feedback controls maintained stability through rapid oscillation of concentrations about equilibrium points, a phenomenon well known in the literature on feedback control. Oscillation of concentration to maintain equilibrium was strikingly illustrated by ATP and ADP concentrations during a shift up from minimal medium to broth (Figures 6, 7). The "restoring force" effectedby the feedback equations enabled the cell to maintain equilibrium concentrations of ATP and ADP. In contrast the concentration of ATP was too high and the concentration of ADP was too low after a simulated shift up without feedback controls in the simulated cell. Similarly an overshoot in ribosomal RNA concentration was quickly corrected by the simulated cell with feedback controls, while a similar simulation experiment on a cell without feedback controls produced so much ribosomal RNA that a real cell would lyse (Figure 10).

9 CONCLUS IONS Preliminary conclusions drawn are 1) Shifts from poor to rich medium are more of a challenge to the cell than shifts from rich to poor medium. The shift up to rich medium requires elaborate feedback control mechanisms, whereas the shift down to poor medium does not require feedback control as strongly, but can be handled by metabolic topology. 2) Oscillations occur about equilibrium concentrations; fixed equilibrium concentrations are not maintained in the simulation. However, this agrees with experimental observations which indicate that real cells are constantly oscillating biochemical systems and suggests, indeed, that the oscillating concentrations produced by feedback control systems are necessary for the flexibility characteristic of living systems.

10 APPENDIX Formally, in constructing a model on which to base a simulation a homomorphic mapping is often used to reduce a complex system description to a relatively simple one (Figure 17). Such a homomorphism is essentially a partition of the state space of the real system which preserves the transition function of the system just as a group homomorphism preserves the group multiplication. In practice, the mapping between real system and model is never truly a homomorphism —indeed when divergence between the behavior of the partitioned real system and the behavior of the model is detected this may initiate a search for more truly homomorphic and hence more adequate models (Ulam, 1966) The kind of partition considered may depend on several factors. The measuring instruments with which experimental observations are made impose certain equivalences which cannot be cut across. The partition must relate to experimental data which is actually or potentially available. It must also be fine enough to maintain distinctions between the parts of the system which are of primary interest. When a simulation by computer is involved, as it is in this paper, additional restrictions are placed on the kind of model that can be considered. The model must not exceed the information processing capabilities of real computers. Suppose for example we momentarily consider describing the state of a biological cell by listing the states of each of the elementary atomic particles of which' the cell is composed. The sheer enormity of the number of such particles would relegate such an approach to the realm of wishful thinking in two ways: one, we would hardly have enough data storage capacity to keeo track of such a long list, and two, we would not have enough program storage capacity to specify how each atomic state changes as a function of the prior states of thoe atoms which

11 influence it. (Moreover, the time required to run such a program would exceed a scientist's patience if not his lifetime). This paper concerns a model of a bacterial cell (E. coli) which has been constructed to enable computer simulation of the cell behavior in its living form. We believe that our model is best understood as an attempt, operating under the constraints on model making just mentioned, to achieve a truly homorphic mapping of a real world system. (We suggest that it is fruitful to deal with the modelling process in general in this way but we do not further argue this proposition.) Accordingly, we devote some time to an exposition of the system theoretic concepts underlying the idea of models as attempted homomorphisms. This development is briefly sketched here and will be more fully expanded in subsequent publications (Weinberg and Zeigler,to be published). In its most basic form, a system is defined as a set of states S, together with a transition function T:S -+ S. T describes the behavior of the system over time by indicating which next state is to follow the present state. Thus if the state at time t is s(t) then the state at time t+l, s(t+l) = T(s(t)). The state space S is usually described as a cartesian product of component state sets i.e. S =x S where D is called a set of co-ordinates (or entities) acD a and S is the state set (or attribute set) of co-ordinate a. In Figure 1 we a list a number of possible state spaces and indicate the form a transition function might take in each. In Figures 2 and 3 we specify in more detail the state space and transition function of the present model. A homomorphism from a system (S, r) to a system (S', r') is a map h from S:onto S' such that for all s e S h(t(s)) = a'(h(s)) Thus, a homomorphism preserves the transition function and guarantees that every

12 state trajectory in (S, T) has a corresponding state trajectory in (S', 1T), As we have stated, model making may be identified as an attempt to obtain a truly homomorphic image of a real system which among other things, is simple enough to implement on a computer. The process of going from lower level to higher level functional units may be viewed in this light. We cannot implement an atomic state model of an E. coli (even though presumably such a simulation would be maximally informative) because such a model would require an information processing capacity well beyond that possessed by any man made computer. At the molecular and concentration levels of Figure 18 the same thing would be true. Notice that each of these levels arises by grouping together co-ordinates at a lower level to form higher level units. It is only by continuing this process one more step that we are able to arrive at models of a cell simple enough to implement on a computer. It is still possible at this level to construct models which can be meaningfully tested against real cellular behavior as consideration of our present simulation has demonstrated. Partitioning the co-ordinates of system to achieve a simpler system can be given a mathematical formulation (Zeigler, to be published). We indicate the conditions under which such a partition will yield a homomorphic image system. We also show how the complexity of the system ( as determined by measures relevant to computer implementation) can be made to decrease in this way. The point to be made, however, is that such a partitioning can be justified mathematically and experimentally at any level, not only the levels — atomic, molecular, etc. —traditionally accepted.

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14 Garfinkel, D. (1966). "A Simulation Study of Mammalian Phosphofructokinase", J. Biol. Chem., 241, 286-294. Gause, G. F. (1966). Microbial bModCls of Cancer Cells, Philadelphia: Saunders. Ginzburg, A. (1968). Algebraic Theory of Automata, New York: Academic Press. Gordon, Geoffrey (1969). System Stimulation, Englewood Cliffs, New Jersey: Prentice-Hall. Griffith, J. S. (1968). "Mathematics of Cellular Control Processes", J. Theoret. Biol., 20, 202-216 Hayes, W. (1968). The Genetics of Bacteria and their Viruses, second edition, New York: Wiley. Heinmets, F. (1964) "Analog Computer Analysis of a Model-System for the Induced Enzyme Synthesis", J. Theoret. Biol., 6, 69-75. Heinmets, F. (1966). Analysis of Normal and Abnormal Cell Growth, New York: Plenum. Holland, J. H. (1969a). "Heirarchical Descriptions, Universal Spaces and Adaptive Systems", in Essays on Cellular Automata (ed. A. W. Burks), Urbana: University of Illinois Press (to appear). Holland, J. H. (1969b). Adaptive Plans Optimal for Payoff only Environments," in Proceedings of the Second Hawaii Conference on System Sciences. IBM Corporation (1969). System/360 Scientific Subroutine Package (360-CM-03X) Version III Programmer's Manual, New York: IBM. Kimura, M. (1965). "Changes of Mean Fitness in Random Mating Populations, when Epistasis and Linkage are Present", Genetics, 51, 349-363. Knuth, Donald E. (1969). The Art of Computer Programming, Volume 2, Seminumerical Algorithms, Massachusetts: Addison-Wesley. Koch, A. L. (1967). "Metabolic Control Through Reflexive Enzyme Action", J. Theoret. Biol., 15, 75-102. Lark, K. G. (1969). "Initiation and Control of DNA Synthesis", Annual Rev. Biochem., 38, 549-604. Maalae, 0. and N. O. Kjeldgaard (1966)." Control of Macromolecular Systhesis, New York: Benjamin. Mahler, H. R. and E. H. Cordes (1966). Biological Chemistry, New York: Harper 6 Row.

15 Mandelstam, J., and K. McQuillen, editors (1968). Biochemistry of Bacterial Growth, New York: Wiley. Mesarovic, M. D., editor (1968). Systems Theory and Biology, New York: Springer-Verlag. Mitchison, J. M. (1969). "Enzyme Synthesis in Synchronous Cultures", Science 165, 657-663. Mize, J. H., and J. G. Cox (1968). Essentials of Simulation, Englewood Cliffs, New Jersey: Prentice-Hall. Moser, H. (1958). The Dynamics of Bacterial Populations Maintained in the Chemostat, Washington, D. C: Carnegie Institution of Washington Publication 614. Murray, A. w. and M. R. Atkinson (1968). "Adosine 5' Phosphorothioate. A Nucleotide Analog that is a Substrate, Competitive Inhibitor, or Regulator of Some Enzymes that Interact with Adenosine 5'-Phosphate", Biochemistry, 7, 4023-4029. Reiner, J. M. (1968). The Organism as an Adaptive Control System, Englewood Cliffs, N. J.: Prentice-Hall. Reiner, J. M. (1969). Behavior of Enzyme Systems, New York: Reinhold. Stahl, W. R. (1967). "A Computer Model of Cellular Self-Reproduction", J. Theoret. Biol., 14, 187-205. Strickberger, M. W. (1968). Genetics, New York: Macmillan. Sugita, M., and N. Fukuda (1968). "Functional Analysis of Chemical Systems in vivo Using a Logical Circuit Equivalent", J. Theoret. Biol., 5, 412-425. Tsanev, R. and B. Sendov (1969). "A Model of Cancer Studied by a Computer', J. Theoret. Biol., 23, 124-234. Ulam, S. M. (1967). in Prospects for Simulation and Simulators of Dynamic Systems, edited by G. Shapiro and M. Rogers, New York: Spartan. Umbarger, H. E. (1969). "Regulation of Amino Acid Metabolism", Annual Rev. Biochem., 38, 323-370. von Neumann, J. (1966). Theorey of Self-Reproducing Automata (edited by A. W. Burks), Urbana: University of Illinois. Waddington, C. H. (1969). Towards a Theoretical Biology, Birmingham, Great. Britain: Kynoch.

16 Wallace, B. (1968). Topics in Population Genetics, New York: W. W. Norton. Walter, Charles (1969a). "Stability of Controlled Biological Systems", J. Theoret. Biol., 23, 23-38. Walter, Charles (1969b). "The Absolute Stability of Certain Types of Controlled Biological Systems", J. Theoret. Biol., 23, 39-52. Weinberg, R. (1968a). "Analytic and Logical Equations in a Computer Simulation of Cell Metabolism and Replication", Sixth Annual Symposium on Biomathematics and Computer Science in the Life Sciences, The University of Texas, pp. 102-103. Weinberg, R. (1968b). "Computer Simulation of a Living Cell", Bacteriological Proceedings, G114. Weinberg, R. (1968c). "Computer Simulation of Self-Reproduction by a Living Cell", Genetics, 60, 235. Weinberg, R. and M. Berkus (1969a). "Computer Simulation of Evolving DNA", Abstract to be published in Biometrics. Weinberg, R. and M. Berkus, (1969b). Computer Simulation of a Living Cell, Ann Arbor, Michigan: University of Michigan Technical Report 01252-2-T. Wendroff, B. (1969). First Principles of Numerical Analysis, Massachusetts: Addison-Wesley. Westley, John (1969). Enzymic Catalysis. New York: Harper & Row. Yeisley, W. G. and E. C. Pollard (1964). "An Analog Computer Study of Differential Equations Concerned with Bacterial Cell Synthesis", J. Theoret. Biol. 7, 485-501.

Figure 1. Natural Groupings. Chemical species in one large block were represented in the simulation as one pool. Figure 2. Flow of Materials. Figure 3. Model of a Living Cell Used for the Computer Simulation. Figure 4. Differential Equations. Quantity to the left of = is the change in amount of the substance; e. g., DDNA represents the change in the amount of DNA in one time increment DT. The differential equation underlying the first equation is DDNA = K(6)*NUC*EK(6)*ATP*DT for a discrete time interval DT. As DT approaches O, we get the underlying continuous differential equation lim D(DNA)/DT = d(DNA)/dt - K(6)*NUC*EK(6)*ATP DT + 0 Figure 5. Logarithm of Various Quantities in a Growing Culture. Logarithmic Increase in Cetl Mass over Time, Stepwise Increase in Number of Cell. Comparative magnitude of various quantities are a function of the scaling factors used in order to plot all quantities on one graph. A cell doubles after a DNA replication cycle. The doubling takes a relatively short time, as indicated by the sudden, stepwise increase in "TOTAL NUMBER OF CELLS", where as "TOTAL DNA" increases throughout the replication cycle. Figure 6. ATP during Simulated Shift Up. Figure 7. ADP during Simulated Shift Up. Figure 8. DNA during Shift Up. WelZ aerated liquid cultures at 370C were used for simulated and laboratory data. Slymbols: 0 irZlated shift from minimal glucose to broth at time zero; C simulated minimal ~ X laboratory shift up; X laboratory minimal glucose.

Figure 9. Protein during Shift Up. Well aerated liquid cultures were used for laboratory and experimental data. Symbols: O simulated shift from minimal glucose to broth at time zero; 0 simulated minimal glucose control; X. laboratory shift up; + laboratory minimal.. Figure 10. Ribosomal RNA during Shift Up. Well aerated liquid cultures at 37~C were used for laboratory and simulated data. Symbols: C simulated minimal glucose liquid medium; 0 simulated shift from broth to minimal glucose at time zero; X laboratory shift up; + laboratory minimal glucose. Figure 11. ATP during Simulated Shift Down. Figure 12. DNA during Shift Down. Well aerated liquid were used for laboratory and experimental data. Symbols: down from broth to minimal glucose at time zero; 0 broth tory shift down; X laboratory control. cultures at 37~C 0 simulated shift control; X labora Figure 13. Ribosomal RNA during Shift Down Well aerated liquid cultures at 370C were used data. Symbols: 0 nutrient broth; 0 shifted borth at time zero; X laboratory shift down; from Broth to Minimal Glucose. for simulated and laboratory to minimal glucose from nutrient X laboratory control. Figure 14. Protein/Cell during Shift Down. Well aerated liquid cultures. at 37~C were used for laboratory and simulated data. Symbols: 0 simulated shift from broth to minimal glucose at time zero; 00simulated broth; + laboratory shift down; X laboratory broth. Figure 15. Simulated Growth in Low Glucose Concentration. WeZl aerated liquid cultures at 37~C were used for simulated and laboratory data. Symbols: all culturies were grown in minimal medium; O laboratory, 40 mg glucose per liter; A laboratory, 4. 10 mg glucose per liter; 0 simulated 4 mg glucose per liter to 4. l mg glucose per liter at time zero 0Q simulater HIGH. Figure 16. Formal Definitions.

Figure 17. Figure 18. Figure 19. Figure 20. by adjustment adjustment of Useful Homomorphisms. Summary of Program. Growth Cycle. Repression and AZZosteric Inhibition. Repression is obtained of KK8K(INTGR). Allosteric inhibition is obtained through C(INTGR). Table 1. Variables in Program.

r - PHENYLALANINE - TRYPTOPHAN TYROSINE SHIKIMIC ACID AMINO!t ACID ( I ~HISTIDINE A i~~ WALL CELL WALL COMPONENTS I./, — -. ICELL MEMBRANE, ------- - - GALACTOSE URt (/L COLYEN URIDINE ~g GLUCOSE WALL PRECURSOR I I RIBULOSE-5-(4 -- GLUCOSE-I- P 4 GLUCOSE 1 GLUCOSE 6 ~ I I GLUCOSE / P / / RIBOSOME P P RIBOSE 5 IPURINE NUCLEOTIDES PYRIMIDINE CLEOTIDES YCIE |PYRIMIDINE NUCLEOT1,_, _. SERINE OROTIC ACID NUCL E- CYSTEINE VALINE ALANINE 4 a el \\ I 3-PHOSPHOGLYCERIC ACID PHOSPHOENOLPYRUVIC ACID ---— \ I GLYCEROLQ PHOSPHOLIPIDS PHOSPHATIDIC ACID. TRIGLYCERIDES (FATS I -t LYSINE ASFGRAGINE AC RPATATF -'V.104 \ + J LEU(:INE / ISOLEUCINE - PYRUVATE ACETYL-CoA - OXALOACETATE r CITRATE \ —2H BUTYRYL-CoA — V. HOMSEFRINF - — MFETHIONINF I - THREONINE 1 -i1 J t \ W.-,. ^A I i oRESPIRATORY CHAIN tEK3 RESPIRATORY ENZYMES ARGININ| GLUTAMINE44 — GLUTAMATE 4PROLINEA -......, -- - KETOGLUTARATE A 10%, a N..., I AMINO ACID GLUCOSE PROTEIN v. [ RIBOSOME figure 1

figure 2

STATE COORDINATE = ENTITY I RANGE OF VALUE OF COORDINATES m ATTRIBUTE OF ENTITY |Pools of Chemicals, PROC(l... PRDC(),, 10) Concentration of pool Enzymes, EK(l),...,EK(O) Concentration of enzyme Messenger RNA, RNK(1),..., RNK(10) Concentration of RNA Genetic Apparatus Amount of DNA, site of replication, number of genes in cell for producing sites for replication Cell Volume Total volume of the cell Cell Number Number of cells represented in the culture TRANSITION FUNCTION (for calculating the state of the system in the next time step from the present state) A. The differential and boolean equations relating concentrations of variables at a given time to the concentrations of those variables DT seconds later. e.g. for AA, the amino acid pool, one needs enzyme EK(2) to catalyze the production of AA. from glucose, and one uses ATP as an energy source. At the same time, AA is lost as it is used for the production of RIB and PRTN. 1. DAA = K(2)*GLUC*DT*EK(2)*ATP - 1.E6/102.*DRIB - (4.E4/102.)*DPRTN production of AA from loss of AA to loss of AA to PRTN GLUC RIB 2. RNK(2) produced EK(2) from AA under the direction of DNA, using ATP for energy. 3. DEK(2) = K(7)*AA(RNK(2)/MRNAO)*DT*EK(7)*ATP 4. RNK(2) itself was produced from NUC under the direction of DNA, catalyzed by. EK8, using ATP for energy. RNK(2) decayed spontaneously at the same time, producing some loss of RNK(2) already present. 5. DRNK(2) " (K8K(2)*NUC*DNA*EK(8)*ATP - KDRNK*RNK(2))*DT production of RNK(2) decay of RNK(2) B. Attosteric modification of enzymes simulated by modifying the rate constant which characterizes all different forms of any enzyme associated with a particular reaction. C. Repression of messenger RNA directing the production of a particular enzyme D. Genetic behavior of DNA in response to.the state of the cell E. Permeabitity figure 3

91 DDNA = K(6)*NUC*DT*EK(6)*ATP DDtNA1 K(6)*(T/DBLE)*.5*1N1*NUC*DT*EK(6)*ATP DDNA2 = K(6)*(T/DBLE)*.5*IN2*NUC*DT*EK(6)*ATP DDNA3 = K(6)*(T/DBLE)*.5*IN3*NUC*DT*EK(6)*ATP C**** 40 MINUTES TO REPLICATE.5 * DNA DO 100 1 = 1,10 100 DRNK(I) = (K8K(I)*NUC*DNA*EK(8)*ATP - KDRNK*RNK(I))*DT C**** UMRNA = DRNK(1) + DRNK(2) + DRNK(3) + DRNK(4) + DRNK(5) + DRNK(6) 1 + DRNK(8) + DRNK(9) + DRNK(10) + DRNK(7) C**** DTRNA = K(10)*NUC*DT*EK(10)*ATP DRIB = K(9)*NUC*AA*DT*EK(9)*ATP DRNA = DMRNA +.25*DTRNA +.75*DRIB DWALL = K(4)*GLUC*DT*EK(4)*ATP C**** DO 101 I = 1,10 101 DEK(I)= K(7)*AA*(RNK(I )/MRNAO)*DT*EK(7)*ATP C**** DPRTN = DEK(1) + DEK(2) + DEK(3) + DEK(4) + DEK(5) + DEK(6) 1 + DEK(8) + DEK(9) + DEK(10) + DEK(7) C**** DNUC= -(2.5E9/660.)*DDNA - (1.E6/660.)*DIMRNA 1 +K(1)*GLUC*DT*EK(1)*ATP - (2.5E4/660.)*DTRNA 2 -(2.E6/660.)*DRIB - K(5)*NUC*DT*EK(5)*ATP C**** DAA=K(2)*GLUC*DT*EK(2)*ATP - 1.E6/102.*DRIB - (4.E4/102.)*DPRTN DATP = K(3)*GLUC*DT *ATP*EK(3) - DNAP*DDNA - MRNAP*DMRNA 1 - TRNAP*DTRNA - RIBP*DRIB - PRTNP*DPRTN-WALLP*DWALL 2 - (AAP*K(2)*GLUC*EK(2)*ATP + NUCP*K(1)*GLUC*EK(1)*ATP +2*K(5)*NUC 3 *EK(5)*ATP )*DT C**** DADP = -DATP + K(5)*NUC*DT*EK(5)*ATP DVOL " K(14)*WALL*DT C**** INCREASE IN VOLUME PER UNIT INCREASE IN CELL WALL Figure 4 Differential Equations: quantity to the left of = is the change in amount of the substance; e.g., DDNA represents the change in the amount of DNA in one time increment DT. The differential equation underlying the first equation is DDNA = K(6)*NUC*EK(6)*ATP*DT for a discrete time interval DT. As DT approaches 0, we get the underlying continuous differential equation lim D(DNA)/DT = d(DNA)/dt = K(6)*NUC*EK(6)*ATP DT -o 0

I.600r —--- 1.600r 7 —----— T —- T- -- I *, TOTAL DNA I I i I I I I I I f ^~o~/ —TOTAL RNA 1.4001- - -I- - -- I I I I I I 1.200- --- T - I I I I 1.000 - - - L I I I 1 I I / -i 1 7LLI LIJ o 0 LJ i _J LL 0 0C 0 -I Z- - _ _1_ I Z y J/ O TOTAL PROTEIN I I I I 1 __ - 1~ 1 - -- E I TOTAL NUMBER I I OF CELLS I I I I! I I 0.800f- - 0.600- /I! S/.I 1 / i 1 4+- -1 -- -t ---- - - - - 1 1 I I I I I 1 L 1 I - - I 1 1 I I 1 I 1 I! I 1 I I I I 1 1 i 0.000 I —0.000 I I I - -, - I m - ~ -_- -1 I 900.000 1800.000 I 2700.000 3600.00C ) I 5400.( 4500.000 TIME IN SECONDS figure 5

LOG2 (ATP MOLECULES/CELL) 20.06 20.00 19.90 19.80 19.70 19.60 - I 19.50 X 19.40 19.30 19.20 19.1019.00 18.90 0 Simulated Minimal to Broth, Feedback -Simulated Minimal, Without Feedback Laboratory Average for Minimal Laboratory Average for X Minimal to Broth -Simulateo Minimal, With Feedback Simulated Minimal to Broth, No Feedback I 1 2 3 4 5 6 7 8 9 10 SECONDS Figure 6. ATP during Simulated Shift Up.

LOG2(ADP MOLECULES/CELL) 18.0 17.5 ~~~~~~~~~~~~~~~~~~',l~q _..(() ~ii jI I,.I:i J~~ Simulated Minimal to Broth, No Feedback Simulated Minimal, No Feedback Simulated Minimal: Feedback 17.0 16.5, 1 16.015.5 15.0 0 Minimal Laboratory Average for Minimal to Broth Simulated Minimal to Broth, Feedback 1 2 3 4 5 6 7 8 9 10 Figure 7. ADP during Simulated Shift Up.

LOG2 (DNA MOLECULES/CELL) 1.00400 1.00350 1.00300 1.00250 1.00200 Simulated Shift up, with Feedbac 0 e-k.'i 0 %L0 A\ Simulated Mineral: Feedback = No Feedback,Laboratory Minimal Laboratory Shift up Simulated Shift up, Without Feedback 1.00100 1.00000 0 1 2 3 4 5 6 SECONDS 7 8 9 10 figure 8

LOG2 (PROTEIN MOLECULES/CELL) 19.470 19.460 19.450 19.440 Simulated Shift up, Without Feedback Laboi Simulated Shift up, Shift 19.430 With Feedback 19.420, La[:./ \ ^-P^~ Min,' Simulated Minimal 19.410 0- 0 / Feedback= No / Feedback 19.4001 — 1 I I I I I i ratory up )oratory imal I 1 2 3 4 5 SECONDS 6 7 8 S 10 figure 9

1339 LOG(RIBOSMAL RNA MOLECULES/CELL) Simulated Shift up, Without Feedback 13.38 13.37 Simulated Shift up, With Feedback 9-.-o- — 0- -0 - / Laboratory Shift up 13.36- / "'X / 1/3 ^ Laboratory Minimal 13.35Simurlated Minimal: Feebcck - =.\ F-edback 13_34 l I I > 0 1 2 3 4 5 6 7 9 10 s E CODnS Figure 10

LOG2 (ATP MOLECULES/CELLS) 21.560 21.500- 4 I: I f1-Simulated.I,1 I1, i ji,Broth,W ith I i J Feedbock 21.300 -- ulte /Shift Down 21.2000lj' t t J l'.|I 21.100_l 21.o000 / I } li.B 20.500 20.430K I 0 Average for 20.600 - U I I'I t, | | Minimalto 1 \i,1! Br oth 20o00* I i:!i J 20.400- 3i i ri I,I i~~~~: 1~~~~~~~i 4 5 6 SECONDS Figure 11.

LOG2 (DNA MOLECULES/CELL) 1.593 _ 1.592 I 591. 1.590 I I Q - Labore Labor( Simuic With F WithoL 1.589 - - atory, Broth = -X atory, Shift Down 3ted Broth::eedback Jt Feedback / /0 2/' / Simulated Shift With Feedback =Without 1.588 R-o I 587 m 1.586 1.585c 0 Feedback I I I I 1 I - - i 1 2 3 4 5 6 7 8 9 10 SECONDS figure 12

LOG2 (RIBOSOMAL RNA MOLECULES/CELL ) 15.446 Laboratory Broth 0. a, ad Mt. _ _ _7 c;,, I. 15.445 15.444 15.443 15.442 15.441 15.440 15.43 9 15.438 15.437 15.436 - 0 ih IIlUFIU edbaUlll With Feedback = Without Feedback Simulated Shift Down: With Feedback = Without Feedback Laboratory Shift Down i 2 3 4 5 6 7 8 9 10 SECONDS figure 13

LOG2 (PROTEIN MOLECULES/CELL) 20,895 20,894 Simulated Broth, With Feedback 20.893 I 20,892 20,891 20,890 Laboratory, Broth Simulated Broth Without Feedbao 20,889 20,888 20,887 20,886 Simulated Shifts Down: Feedback = No Feedback Laboratory Shift 20,885 20,8840 Down 1 2:3 4 5 6 7. 9 figure 14

13.348 0 LABORATORY o-o HIGH / S!MUL j HIGH I 0 %, 13.347 / C,) w D O L/ LABORATORY o I LOW = 13.346 SIMUSIlMULA < E - / 0^ SHIFT TO LO3 0 CO ( 0 13.345 _J 1 2 3 4 5 G SECONDS 7 S 9 10 figure 15

(S, T) a ys.tem where S - 4tate 4pace T: S + S (toanCtion fanction) t+ I - T (%t) S, given a" a cata^sin product S - x S oeP a wheAe D - 6et o co-ordinate (entitie) S - 4tate 64et (attcibute et:) o0 co-odinate a Homomotphihm h fdom (S. T) to (S' T') i6 a map h:S -S' 46uch thatt dot aU 46eS h(T(4)) - T'(h(4)) Homomoaphism4 may be obtained by paAtitontng co-otdinate 4 Pe D 4uch fthat the blocks, ot. higheA teveZ co-otdinate p4r4eAeve the tAcnition 6fnction. figure 16

* ------------- 7-I — - -------— ~ —--- -I I — STATE SPACE CO-ORDINATES CO-ORDINATE SETS TRANSITION FUNCTION Atomic Electrons, Position, Schrodinge nuclei momentum m an (quantum mechanics) Molecular Molecules Shape, active site, energy level e.g. Koch probabilistic model Mathematical and Concentration Molecule type Number of molecules logical equations e.g. ADP, DNA of ADP/cell based on chemical etc. kinetics e.g. Chance et al.~~~~~~~~~~~~~~~~~~~~~~~ Higher level'cI Ioings Bio-chemical pools e.g. amino acids Number of molecules of pool/cell Mathematics and logical equations based on chemical kinetics and Molecular control mechanisms e.g. repression, allosteric inhibition (Weinberg, Goodwin) Figure 17.

TIME = 0 r tl CALCULATE WHICH AND ADJUST RATE TION UNDER THIS MRNA MOLECULES ARE REPRESSED CONSTANTS FOR THEIR FORMACONDITION. MODIFY ENZYME CONSTANTS FOR ALLOSTERIC INHIBITION IF APPLICABLE. CREATE THE SITE OF REPLICATION FOR NEW CHROMOSOME IF ENOUGH INITIATOR IS AVAILABLE. I,,,,,...iii i i, i l, CALCULATE THE CONTROLS ON THE INITIATOR PRODUCTION, THE NUMBER OF GENES IN THE PRODUCTION, AND ANY NEW INITIATOR PRODUCED. PRODUCE MRNA, TRNA AND THE RESULTING ENZYME PRODUCTS. _ _ _,,,,,,,,, i i CALCULATE THE NEW CONCENTRATIONS FOR THE ADJUSTMENT IiN VOLUME. (THESE'SHOULDN'T CHAN'GE.) CALCULATE THE INCREASED POOL VOLUM~ES. CHECK TO SEE THAT RATIO BETWEEN CONCENTRATION OF POOLS AND THE BASE LEVELS EQUALS ONE. IF MORE THAN ONE CHROMOSOME HAS REPLICATED AND NO CHROMOSOME IS INCOMPLETELY REPLICATED, DIVIDE AND PRODUCE TWO NEW CELLS. TIME = TIME + DT Increment Time Counter l~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ TIME > LIMI, Yes CONTINUE Figure 18. Swmmary of Program.

FEED IN INTERNAL CELL CONCENTRATIONS (IN NUMBER OF MOLECULES PER CELL) FOR ENVIRONMENT (I). 1 = MINIMAL MEDIUM, 2 - AMINO ACIDS, 3 = BROTIH. CALCULATUE PRELIMIN&ARY F7LO;,' RASIA'A CONSTANTS FOR FLOW OF MATERIAL FROM ONE POOL TO ANOTHiER e.g., CHANGE IN DNA PER TIME INCREMENT = K6*DNA. CALCULATE ENZYME RATE CONSTANTS, REPRESSOR LEVELS, REPLICATION ROUTINE IN ORDER TO FIT EXPERIMENTAL DATA. CALCULATIONS BY GROWING CELL IN ENVIRONMENT (I) FOR A FEW GROWTH CYCLES, NOT YET USING ALLOSTERIC INHIBITION TO ACHIEVE PROPER RATES OF CATALYSIS. STORE VALUES CALCULATED, AND WHICII.ENVIRONMENT TH-EY WERE CALCULATED FOR. no rI 3 ----- I 1 I f 1 yes CALCULATE ALLOSTERIC MODIFICATION OF ENZYMES NECESSARY TO GIVE OBSERVED GROWTH RATES IN THE THREE ENVIRONMENTS INVESTIGATED. GROW CELL FOR SEVERAL GROWTH CYCLES IN ENVIRONMENT (I). LET CELL ADJUST TO ENVIRONMENT I BY USING REPRESSION, REPLICON CONTROL, AND ALLOSTERIC INHIBITION OF ENZYME ACTION. IF CELL CAN ADJUST TO DIFFERENT ENVIRONMENTS, THE CALCULATIONS ARE CONSIDERED A PRELIMINARY SUCCESS. ENDno yes figure 19

X = 1./(1. -.09*R(1) -.40*R(2) -.40*R(3) -.05*R(4) l -.01*R(5)) X = X*MRNA DO 10 II = 1,10 K8K(II) = K(8)*RC(II)*(1 - R(II))*X/MRNAO Above Three Equations Effect Repression P1 = P1V(II) P3 = P3V(II) C(II) = ( KK(II) + P1*KB(II)*PRDC K(II) 1 + P3*KBB(II)*PRDC K(II) **2)/(1 + P1*RPDC K(II) 2 + P3*PRDC K(II) **2 Above Three Equations Effect Allosteric Inhibition Figure 20 Repression and Allosteric Inhibition. Repression is obtained by adjustment

TABLE 1 VARIABLES IN PROGRAM: A = array 0 = floating point 1 = integer A2 A3 AAO AAP AAO ADJST ADPO ADP ATPO ATP ATPSB BROTH CAA CHRMO CNTRL COUNT CRAZY C(I) DAA DADP A 0 arrays used in solve function to obtain rate constants used A 0 in allosteric inhibition O amino acid concentration at time zero O ATP molecules used to make 1 amino acid molecule O amino acid concentration O adjustment factor for concentrations from volume increase O ADP concentration at time 0 O ADP concentration O ATP concentration at time zero O ATP concentration A 0 array to store ATP concentrations in different environments 1 equals 1, if, cell growing in broth 1 equals 1 if cell growing in casamino acid O number of chromosomes at time 0 1 equals 1 if cell using metabolic controls to adjust growth rate O number of growth cycles made 1 used as a logical variable A 0 enzyme rate constants.0 change in amino acid concentration O change in ADP concentration DAP02 DATP DDNA1 DDNA2 DDNA3 DDNA DEK(1) DEK(2) DEK(3) DEK(4) DEK(S) DEK(6) DEK(7) DEK (8) DEK(9) O change in O change in in one O change in O change in O change in ATP concentration from literature ATP concentration calculated from rate cdnstants time step amount chromosome 1 in one time increment amount of chromosome 2 in one time increment amount of chromosome 3 in one time increment 0 change in total DNA in one time increment O change in enzymes for nucleotide production in one time increment O change in enzymes for amino acid production in one time increment O change in enzymes for glycolysis production in one time increment O change in enzymes for wall production in one time increment O change in enzymes O change in enzymes O change in enzymes O change in enzymes 0 change in enzymes for ADP, ATP synthesis in one time increment for DNA sy;nthesis in one time increment for protein production in one time increment for MINA synthesis in one time increment for ribosome synthesis in one'time increment

DEK(10) 0 change in enzymes for TRNA production in one time increment DIN 0 change in initiator concentration in one time increment DNAO 0 DNA at time zero DNA1 0 chromosome 1 "concentration", i.e., amount/volume of cell DNA1Z 0 chromosome 1 at zero time DNA1T 0 total chromosome 1 DNA2 0 chromosome 2 "concentration" DNA2T 0 total chromosome 2 DNA2Z 0 chromosome 2 at zero time DNA3 0 chromosome 3 "concentration" DNA3T 0 total chromosome 3 DNA3Z 0 chromosome 3 at zero time DNAP 0 ATP used per DNA molecule synthesized DNA 0 DNA DNASB A O array to save concentrations of DNA in different environments DNUC O change in nucleotide concentration DBLE O time for cell to go through one reproductive cycle DPRTN O change in protein in one time increment DRIB O change in ribosome in one time increment DMRNA O change in MRNA in one time increment DRNA O change in total RNA in one time increment DRNK(i) A O change in MRNA for enzyme EK(i) in one time increment. i ranges from 1 to 10. DT O length of one time increment, = differential DUM1 0 dummy variable in solve function DUM2 0 dummy variable in solve function DUM3 0 dummy variable in solve function DVOL O change in cell volume in one time increment DWALL O change in cell membrane and cell wall in one time increment DPRDK A O array of change in product concentration in one time increment PRD A O the stored array of the previous four product values, for predictor corrector DPRD A O array of the four previous D(product) values for the predictor corrector PPRD A. 0 current array of the predictor values of products CPRD A O current array of corrector values of products EK(1) 0 concentration of enzymes for nucleotide production EKZ(1) O concentration of enzymes for nucleotide production at zero time EK(2) 0 concentration of enzymes for amino acid production EKZ(2) 0 concentration of enzymes for amino acid production at zero time

where 3 indicates 4 indicates 5 indicates 6 indicates 7 indicates 8 indicates 9 indicates 10 indicates glycolysis cell wall production ADP, ATP production DNA production protein production MRNA production ribosome production TRNA production FACTR GLUCO GLUC ID IN11 O factor by which chromosomes multiply in one reproductive cycle O glucose concentration at zero time O glucose concentration 1 integer variable in RPLACE routine O site for replication of chromosome 11, = 1 if it is present INllZ IN1 IN1Z IN21 IN21Z O site O site O site O site O site for for for for for for for for for for replication replication replication replication replication replication replication replication replication replication of chromosome 11 at zero time of chromosome 1, = 1 if it is present of chromosome 1 at zero time of chromosome 21 of chromosome 21 at zero time of chromosome 2 of chromosome 2 at zero time of chromosome 31 of chromosome 31 at zero time of chromosome 3 IN2 IN2Z IN31 IN31Z IN3 0 site 0 site 0 site 0 site 0 site IN3Z IN II INZ K(1) 0 site for replication of chromosome 3 at zero time O concentration of initiator in cytoplasm 1 an integer variable O initiator concentration at zero time O preliminary rate constant for nucleotide production K(2) K(3) K(4) K(5) K(6) O preliminary O preliminary O preliminary..0 preliminary O preliminary O preliminary O preliminary O preliminary O preliminary O preliminary of wall rate rate rate rate rate rate rate rate rate rate constant constant constant constant constant constant constant constant constant constant for for for for for for for for for for amino acid production glycolysis cell wall production ADP production DNA production protein production MRNA production ribosome production TRNA production volume increase as a function K(7) K(8) K(9) K(10) K(14) KDRNK K8K(i) K8KZ(i) KBB(i) KB 0 rate constant for MRNA decay A 0 rate constant for MRNA EK(i) A 0 rate constant for MRNA for EKZ(i) A O rate constant for allosterically inhibited enzyme EK(i) with two molecules of product attached to the enzyme A O array of rate constants of allosterically inhibited enzymes with one molecule of product attached to the enzyme

KIN 0 preliminary rate constant for initiator production K8K(i) A 0 rate constant for production of EK(i) KK(i) A 0 rate constant for uninhibited enzyme EK(i) K(i) A 0 array to store preliminary rate constants, used for each environment LN2 0 natural logarithm of 2 L 1 integer variable for calling on solve function M 1 integer variable for printing loop MRNAO 0 MRNA concentration at time zero MRjNAP 0 ATP per MRINA molecule produced MRNA 0 MRNA concentration MULT 0 number of genes producing initiator NO 0 number of cell in population (doubles when cell divides) NUCO 0 molecules of nucleotide at zero time NUCP 0 molecules of ATP to make one nucleotide NUC O concentration of nucleotide P1 0 rate constant P1V A 0 array of equilibrium rate constants for enzymes P3 0 equilibrium rate constant for two molecule allosteric inhibition P3V A 0 array of equilibrium rate constants for two molecule allosteric inhibition PRDCO A 0 array equivalenced to products at zero time PRDCK A 0 array equivalenced to products PRDC A 0 array for storing concentrations of products in different environments PRDC(1) 0 NUC PRDC(2) 0 AA PRDC(3) 0 ATP PRDC(4) 0 WALL PRDC(5) 0 ADP PRDC(6) 0 DNA PRDC(7) 0 PRTN PRDC(8) 0 MRNA PRDC(9) 0 RIB PRDC(IO) 0 TRNA PRDC(11) 0 GLUC PRDC(14) 0 VOL PRTNO 0 protein concentration at zero time PRTNP 0 ATP molecules used per protein molecule formed PRTN 0 proteini concentration RAA 0 ratio of aaino acid concenzration to a base level RADP 0 ratio of AD? conceP,,ontr.: D~o to a base clvel RATP 0 ratio of ATP concentration to a base level

RC RDNA1 RDNA2 RDNA REK(i) A 0 array O ratio O ratio O ratio A O ratio of repression constants for MRNA repression of chromosome 1 concentration to a base level of chromosome 2 concentration to a base level of DNA concentration to a base level of EK(i) concentration to a base level, i = 1,...,10 RI BO RIBP RIB RNAO RNA TRNAO TRNAP TRNA RNK(i) RNKZ(i) RNUC RON RPRTN RRIB RMRNA O ribosome concentration at time zero O ATP used per ribosome made O ribosome concentration O RNA concentration at time zero O RNA concentration O transfer RNA concentration at time zero O ATP per transfer RNA molecule made 0 transfer RNA concentration A 0 concentration of MRNA for enzyme EK(i), i = 1,...,10 A 0 concentration at zero time of MRNA for EKZ(i), i = 1,...,10 O ratio of nucleotide concentration to a base level 1 used as a logical variable turning repression on O ratio of protein concentration to a base level O ratio of ribosome concentration to a base level O ratio of MRNA concentration to a base level RRNA RTRNA RRNK (i) R RVOL RWALL SUM T VOLO VOLN O ratio O ratio A O ratio A 0 array O ratio of RNA concentration to a base level of TRNA concentration to a base level of IRNK(i) concentration to a base level, for repression constants of new volume to old volume at end of one i = 1,...,10 time increment O ratio of pool for wall to a base level in terms of concentration A 0 array used in solve function O generation time in seconds O volume of cell at time zero O volume at end of one time increment VOL WALLO WALLP WALL X XK(i,j) XEK(i,j) XKS(i,j) O volume O concentration O ATP molecules O concentration O variable used of pool for wall production at time zero used per molecule of cell wall produced of pool for wall production in repression routine A 0 value of K(i) in environment (j) A O value of EK(k) in environment (j) A O value of K8K(i) in environment (j)

ACKNOWLEDGEMENT This work was supported bya Public Health Service Fellowship 2-F3-CA-6931-03 from the National Cancer Institute awarded to the senior author; Public Health Service Grant GM-12236 from the National Institute of Health, Army Research Office-Durham Contract DA-31-124-APO-D-483 and Office of Naval Research Contract N00014-67-A-0181-0011.

INC ASST FT.ED Security Classification r DOCUMENT CONTROL DATA * R&D (Security claeeification of title, body of abstract and indexing annotation must be entered when the overall report is claueilied) 1 1. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY C LASSIFICATION LOGIC OF COMPUTERS GROUP Unclassified The University of Michigan 2b GROUP Ann Arbhor Michigin-n.. 3. REPORT TITLE COMPUTER SIMULATION OF A LIVING CELL: MULTILEVEL CONTROL SYSTEMS 4. DESCRIPTIVE NOTES (Type of report and inclusive datee) Technical Report S. AUTHOR(S) Last name. fiint name, initial) Weinberg, Roger Zeigler, Bernard P. 6. REPORT DATE 7a. TOTAL NO. OF PAGES 76. NO. OF rEs December 1969 48 59 ea. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) DA-31-124-ARO-D-483 b. PROJECT NO. c. O* b. OTHIR REPORT NO(S) (Any other numbere that may be asaigned #thi report) d.'10. A V A IL ABILITY/LIMITAtION NOTICES Distribution of This Document is Unlimited 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U. S. Army Research Office (Durham) Durham, North Carolina 13. ABSTRACT A simple bacterial cell (Escherichia coli) has been modeled, and the input-output behavior of the model has been simulated as a program in FORTRAN IV for an IBM 360/67 digital computer. Automata theoretic analysis of the homomorphic model underlying the computer simulation enables us to investigate the information content and complexity of the simulation and of the measurement space representing our data base. The simulated cell is able to adjust its enzymes and DNA to grow in different chemical environments using allosteric modification of enzymes, and repression of RNA synthesis. It grows at realistic rates and achieves limited metabolic stability. DD IJAN, 4 1473 UNCLASSIFIED Security Classification

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