An Agent-based Model of the Lactose Operon

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1 Biomolecular Swarms An Agent-based Model of the Lactose Operon Christian Jacob Dept. of Computer Science, Faculty of Science Dept. of Biochemistry & Molecular Biology, Faculty of Medicine University of Calgary, Calgary, Alberta, CANADA Ian Burleigh Dept. of Computer Science, Faculty of Science University of Calgary, Calgary, Alberta, CANADA Abstract. We present our latest version of a swarm-based, 3-dimensional model of the lactose (lac) operon gene regulatory system. The lac operon is a well-understood genetic switch capable of self-regulation dependent on the energy source of lactose. Our model includes a 3D visualization which simulates proteins as agents with physical properties that interact with DNA, molecules, and other proteins, incorporating many of the important aspects of a genetic regulatory system. Our model utilizes a decentralized swarm approach with multiple agents acting independently according to local interaction rules to exhibit complex emergent behaviours, which constitute the externally observable and measurable switching behaviour. 1 Keywords: Agent-based Biological Modelling, Gene Regulatory System, Lactose Operon, Bioinformatics, Simulation, Swarm Intelligence 1. Introduction Current research in genomics focuses on understanding the genetics of model organisms, such as the bacterium Escherichia coli, the nematode Caenorhabditis elegans, and the fruitfly Drosophila melanogaster. Working with these simple biological models helps to elucidate more complex processes found in higher order gene networks. Major advances in systems biology will more and more be enabled by the utilization of computers as an integral research tool, leading to new interdisciplinary fields within bioinformatics and biological computing. Innovations in agent-based modelling, computer graphics and specialized visualization technology, such as the CAVE R Automated Virtual Environment, provide biologists with unprecedented tools for research in virtual laboratories (Burleigh et al., 2003). In this paper, we present our latest version of a swarm-based, 3D model of the lactose (short: lac) operon, one of the most basic and well-understood biological systems of gene regulation. Several computer-based models of the lac operon exist, including simple grammar-based approaches (Collado-Vides, 1992), 1 to appear in: Journal of Natural Computing, Kluwer, c 2004 Kluwer Academic Publishers. Printed in the Netherlands. LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.1

2 2 Jacob and Burleigh Functional Hybrid Petri Net models (Matsuno et al., 2001), systems based on rewrite rules (Suen and Jacob, 2003), and systems based on large sets of differential equations (Tomita et al., 2000). However, current models of biomolecular systems, such as the lac operon, still have major shortcomings regarding their usability for biological and medical research. Most models do not explicitly take into account that the measurable and observable dynamics of biomolecular systems result from the interaction of a (usually large) number of agents, such as proteins, peptides, signaling molecules or macromolecules (e.g., DNA). With our agent-based models, simulations and visualizations that introduce swarm intelligence algorithms into biomolecular systems, we develop highly visual, adaptive and user-friendly innovative research tools, which, we think, will gain a much broader acceptance in the biological and life sciences research community thus complementing most of the current, more abstract mathematical and computational models (Salzberg et al., 1998), (Bower and Bolouri, 2001). In this paper we propose an agent-based model of the lac operon that incorporates many important gene regulatory aspects of the system in a spatial, 3-dimensional cell environment, including the more universal processes of transcription and translation. In section 2, we present a brief synopsis of the lac operon gene regulatory system as it is commonly understood in biology. In section 3, we discuss our agent/swarm-based implementation of the lac operon, highlighting the modelled processes and structures. Section 4 gives a step-by-step description of a simulated lac operon switching cycle, which we analyze in more detail in Section 5, showing the validity of our model. 2. The lac Operon: A Gene Regulatory System An operon is a group of genes located on the DNA of bacteria. Jacques Monod and François Jacob first studied the lac operon in the 1960s (Jacob and Monod, 1961). Found in the bacterium Escherichia coli (E. coli), the lac operon paradigm stands as a key finding in genetics, as it constitutes one of the most basic gene regulatory systems known, and is consequently used as a basis for studies of more complex genetic systems. The lac operon, in particular, is a gene system that is responsible for converting the sugar lactose into glucose, a key energy source for the bacterium, and galactose. E. coli is a prokaryotic organism without a nucleus that is normally found in a lactose-rich environment, such as the gut of humans. E. coli requires glucose for much of its growth LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.2

3 An Agent-based Gene Regulation Model 3 and has evolved a solution for obtaining glucose from its environment by converting lactose into glucose and galactose. This conversion is accomplished through the enzyme β-galactosidase, which is one of the products of the lac operon. In the presence of lactose, the lac operon is turned on and hence produces β-galactosidase. When lactose is no longer present, the lac operon turns itself off and consequently stops the production of β-galactosidase, thus conserving cellular resources. In this manner, the lac operon is capable of sophisticated self-regulation mainly mediated by the interactions of repressor proteins, lactose, β- galactosidase, and the DNA (Beckwith and Zipser, 1970), (Müller-Hill, 1996), (Ptashne and Gann, 2002) Self-Regulation of the lac Operon Gene-based self-regulation is an emergent property resulting from the interaction of proteins, enzymes, molecules, and DNA. In order to understand how this emergence is accomplished through the interactions of swarms of agents (on which our simulation is based), we will describe the lactose operon in much closer detail (Figs. 1 & 2). The main components of the lac operon as a regulatory unit on the bacterial DNA consists of four genes: lacz, lacy, laca, and laci Gene Complex 1: lacz-y-a The lacz-y-a genes appear as a single module and are located adjacent to one another on the operon (Fig. 1). A control complex consisting of an operator and a promoter precedes the three genes. The operator controls the expression of these genes. Producing a protein from a given gene is accomplished through RNA polymerase, which reads a sequence of genes, resulting in the production of their corresponding proteins through the processes of transcription and translation (Section 2.2) Gene Complex 2: laci The laci gene, the second key module, is located downstream of the main lac complex (Fig. 1). It likewise contains a promoter region, and produces proteins with the help of RNA polymerase. The laci gene product is known as a repressor, which has the ability to bind to the operator region and prevent RNA polymerase from reading the lacz- Y-A genes. Hence, the repressor serves as the basic control mechanism for the lac operon. LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.3

4 4 Jacob and Burleigh RNA Polymerase LacI P i P O LacZ LacY LacA I Repressor binds to operator mrna + Ribosomes No mrna and no proteins I Figure 1. After RNA polymerase docks onto P i, the LacI promoter site, it transcribes the LacI gene into its mrna representation, which is then translated by ribosomes into the repressor protein I. This repressor binds to the LacZ-Y-A operator site, which in turn blocks RNA polymerase; hence, none of the three genes are expressed Turning the Switch When lactose enters the cell, the lac operon can turn itself on (Fig. 2). This is accomplished through the binding of lactose to the repressor to form a repressor-lactose complex. Due to conformational changes, the repressor-lactose complex cannot bind to the operator region of the lacz-y-a genes any more. Consequently, this allows RNA polymerase to now read lacz, lacy, and laca producing β-galactosidase, lactose permease, and transacetylase, respectively. Among these three gene products, β-galactosidase is the enzyme that converts lactose into glucose and galactose. Lactose permease enhances the movement of lactose from the outer environment into the cell, whereas transacetylase does not seem to play a role in this regulatory system (Ptashne and Gann, 2002), (Alberts et al., 1998). Once lactose is removed from the system, the repressor is, again, free to bind to the operator region and terminate the production of β- galactosidase (Fig. 1). In this manner, the lac operon is able to regulate its own gene products. LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.4

5 An Agent-based Gene Regulation Model 5 RNA Polymerase LacI P i P O LacZ LacY LacA mrna + Ribosomes blocked mrnas + Ribosomes Lactose I Z Y A Conformational change Figure 2. When lactose enters into the cell, it induces a shape change in the repressors that disables them from binding to the operator. Consequently, the LacZ-Y-A genes are accessible by the RNA polymerase and are expressed as proteins Z, Y, and A Transcription and Translation Once genes are switched on, RNA polymerase has access to the encoding regions of the structural genes on the DNA. The processes of transcription and translation serve as intermediary steps in order to produce proteins from a given gene. Transcription is the process of converting Deoxyribose Nucleic Acid (DNA) into an intermediate molecule known as messenger Ribonucleic Acid (mrna). The enzyme RNA polymerase is responsible for this particular conversion, which proceeds as follows: (1) RNA polymerase searches along the DNA structure until it encounters an appropriate promoter region. (2) Starting at the promoter region, RNA polymerase begins to synthesize mrna based on the genes found adjacent to the promoter. (3) Once transcription is complete, the mrna strand is free to undergo a second conversion process (through translation), whereas RNA polymerase reiterates the process of transcription. During translation a protein is synthesized from an mrna strand. This mrna-to-protein conversion is achieved through the action of ribosomes and transfer RNA (trna) as follows: (1) A ribosome locates and attaches to a free mrna strand. (2) The ribosome begins to read LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.5

6 6 Jacob and Burleigh the strand and synthesizes a chain of amino acids with the support of trna. The chain then folds into a 3-dimensional protein structure. Multiple ribosomes can simultaneously read and synthesize proteins from a single mrna strand. Once translation is complete, the ribosome detaches from the mrna strand and releases the newly made protein. Figure 3. Zooming into a simulated E. coli cell. All intra-cellular interactions are confined within a spherical cell. The cell wall is being opened while getting closer towards the center of the cell. 3. A Biomolecular Swarm Model Our computer implementation of the lactose operon model and its visualization incorporates a swarm-based approach with a 3D visualization (Fig. 3) (Bonabeau et al., 1999). Each individual element in the simulation is treated as an independent agent governed by simple rules of interaction. Dynamic elements in the system move randomly, executing specific actions when interacting with other agents, which all operate within the confines of the cell. Each agent follows a set of rules that define its actions in the system. As an example, we show a sample of the behaviour rules of RNA polymerase in Table I. The simulation system provides each agent with basic services, such as the ability to move, rotate, and determine the presence and position of other agents. A scheduler implements time slicing by invoking each agent s Iterate method, which executes a specific, context-dependent action. These actions are based on the agent s current state, and the LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.6

7 An Agent-based Gene Regulation Model 7 state of other agents in its vicinity. Our simulated agents work in a decentralized fashion with no central control unit to govern the interactions of the biomolecular agents. There are two specific instances where we have restricted the randomwalk movements of agents (Allen, 2003) in order to more acurately capture agent interactions with the DNA. RNA polymerase has a natural affinity for DNA. Hence, our RNA polymerase agents will randomly move within a defined area located around the DNA. In addition, repressor proteins have a high affinity for the operator region of the lac operon. Consequently, we direct the repressors towards the operator region, while maintaining their random movements. A swarm-based approach affords a measure of modularity, as agents can be added and removed from the system, producing different results each time the simulation is run. This is in contrast to common models of gene regulatory systems that are usually scripted. In addition, completely new agents can be introduced into the simulation. This allows for further aspects of the lac operon to be modelled, such as lactose permease or the CAP activator complex that promotes the production of β-galactosidase Modelling Circular DNA We represent the actual encoding of the lactose operon gene as a circular DNA double-helix with its characteristic Watson-Crick complementarity pattern (Figs. 4 & 5) 1 (Watson and Crick, 1953). DNA consists of four nucleotide bases: Adenine, Cytosine, Guanine, and Thymine. A grouping of three such bases is known as a codon, which codes for a specific amino acid, the basic building blocks of proteins. Due to the vast amount of bases that make up the genes involved in the lac operon, we represent the genetic information as codons. These codons directly correspond to the amino acid composition of their associated proteins. We visualize these codons as colour-coded cylindrical sections that make up the DNA strand (Fig. 5) Modelling Gene Structures There are two distinct gene regions in the lac operon: the laci and the lacz-lacy-laca region (compare Section 2.1). For the purposes of this model, we have chosen to only model the laci and lacz gene regions. The lacy and laca genes do not greatly impact the understanding 1 The DNA is kept still within the cell. In this model, we do not consider any thermal fluctuation of DNA, such as translation, rotation, or chain flexibility. LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.7

8 8 Jacob and Burleigh Table I. Rules governing the behaviour of RNA polymerase as an example swarm agent. Pseudocode is presented with each state of RNA polymerase outlined. The corresponding biological actions are described in the right column. Iterate Pseudo Code Biological State and Action case state of FLOATING: /* initial state */ if near DNA: attach to nearest DNA codon state = DOCKED else: move randomly within the cell DOCKED: if promoter region is reached: state = READY TO TRANSCRIBE else: move along DNA to next codon READY TO TRANSCRIBE: create an empty mrna molecule state = TRANSCRIBING TRANSCRIBING: if a stop codon is reached: release constructed mrna state = DETACHED else if blocked by a repressor: destroy partial mrna state = DETACHED else: move to the next codon append codon mrna DETACHED: detach self from DNA move randomly state = FLOATING Floating: RNA polymerase is usually found near DNA and moves about the cell in a random manner. In this state, RNA polymerase will attempt to attach itself to the nearest free DNA strand. Docked: Once RNA polymerase has docked onto a free DNA strand, it will begin reading the DNA. Ready to Transcribe: When a promoter/operator sequence is found, the RNA polymerase will begin to initiate transcription. Transcribing: RNA polymerase will transcribe the DNA sequence into an mrna molecule. RNA polymerase reads each codon sequentially, and appends a new base to the growing mrna molecule. This process is completed once RNA polymerase encounters the appropriate stop codon. RNA polymerase will then detach itself from the DNA. Detached: Once RNA polymerase has detached from DNA, it will again resume its random movement in the cell. end case LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.8

9 An Agent-based Gene Regulation Model 9 Figure 4. RNA polymerases attach to the DNA and start scanning along a strand. Once a lac operon is identified, the polymerases search for a viable promoter region to begin transcription. of the system and are therefore not included in our current model. 2 The laci and lacz gene regions are labeled appropriately on the model (Fig. 5). In addition, the promoter and operator regions that precede the lacz gene are also included. To further clarify the model, the codon numbering is shown as well, highlighting various aspects of the DNA coding sequences. Conventional models of DNA include the 10 TATA box and the 35 TTGACA RNA polymerase recognition sites, relative to the promoter region. 2 The codons around the two operator sites and the stop codons represent the actual sequences from the E. coli genome. The rest of the circular DNA consists of random codons. Incorporation of the complete lac operon-related genome is possible in our model and will be part of a next version of our biomolecular simulation system currently under construction. LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.9

10 10 Jacob and Burleigh Figure 5. The operator region and the lacz gene on the double helix. Each strand is composed of colour-encoded codons. Shown are also the 10 TATA box and the 35 TTGACA RNA polymerase recognition sites. Analogous labels exist for the laci gene region. Figure 6. Once transcription is initiated (Fig. 4), RNA polymerases produce mrna strands, undergoing translation by multiple ribosomes. The ribosomes construct the amino acid (AA) chains of unfolded proteins (repressors and β-galactosidases) based on the mrna codon sequence. LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.10

11 An Agent-based Gene Regulation Model 11 Figure 7. The repressor binds to the operator region of the lac operon and turns it off. A lactose-repressor complex has formed, preventing the repressor protein from binding to the operator Modelling Transcription and Translation RNA polymerase, the initiator of transcription, is represented as a dark brown (detached) or pink (attached) sphere (Fig. 4). Once RNA polymerase attaches to a DNA region, it starts scanning along the chain of codons. Transcription occurs once RNA polymerase has encountered a viable promoter region. Genes adjacent to the promoter region are transcribed into mrna, represented as a twisted single-strand helix (Fig. 6). Again, we have taken the liberty of representing the mrna gene material as codons corresponding to the actual nucleotide base sequence. The process of translation occurs once the mrna strand has been synthesized. Ribosomes attach to a free mrna strand and begin to synthesize the associated protein. The unfolded protein is shown as a strand of disks. Multiple ribosomes can simultaneously read a single mrna strand, as illustrated in Figure 6. Once a chain of amino acids is completely synthesized, the unfolded protein turns into its associated protein, such as a repressor or β-galactosidase. All folded proteins are represented as spheres of different sizes and colours, more details of which are described in the following section. LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.11

12 12 Jacob and Burleigh (a) (b) (c) (d) (e) (f) Figure 8. Different stages of the lac operon simulation. (a) RNA polymerase searches for a promoter region. (b) RNA polymerases synthesize mrna molecules. Ribosomes synthesize proteins. (c) Repressors (on the bottom right) around the operator block RNA polymerase from transcribing the LacZ gene. (d) Lactose is introduced into the system. (e) Lactose binds to repressors preventing them from blocking RNA polymerase. Three RNA polymerases (on the left) have just started transcribing the LacZ gene. (f) Most of the lactose is split into glucose and galactose. LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.12

13 An Agent-based Gene Regulation Model Simulating Gene Regulation In the case of the lac operon, the two kinds of proteins synthesized through the processes of transcription and translation are repressor proteins and β-galactosidase enzymes (Figs. 7 & 8). Repressors have a natural affinity for the operator region of the lac operon. They attempt to bind to the operator region and physically block transcription of the lacz gene. This turns the lac operon off. This sequence of events is illustrated in Figure 8(a-c) through snapshots taken during our simulation over 1000 iteration steps. In Figure 8c the operator site is surrounded by a number of repressors, which ensure that the operator is blocked (almost) all the time, such that no RNA polymerase can proceed past the operator site. 3 Therefore, at this stage, the expression of β-galactosidase is suppressed, whereas repressors are still produced (see the mrna strands and ribosomes working in the background on the right half of the DNA). Once lactose is introduced into the cell (Fig. 8d), two things will happen. First, repressor-lactose complexes are formed, which cause any bound repressor to be released from the operator site (Fig. 7). This, in turn, enables RNA polymerases to pass beyond the operator and initiate expression of β-galactosidase. In Fig. 8e, three polymerases have already started to scan past the operator in the bottom left part of the DNA. Second, each of the produced β-galactosidases will start to break down lactose into glucose and galactose (Fig. 8f). As soon as all lactoses, including those bound to any repressor, are broken down, repressors will again start to attach to the lacz operator, blocking any further production of β-galactosidase. All the particles (except RNA polymerase and ribosomes) in the simulation system have a predefined lifespan, so that if a protein is not constantly expressed, it will eventually be degraded. Consequently, the simulated cell will finally switch back to a state analogous to Fig. 8c, where only repressors are expressed. 5. Analysis of Simulation Data During each simulation we protocol the concentrations of all particles involved. Figure 9 shows the concentration graphs obtained from the simulation illustrated in Figure 8, which ran over 1000 iteration steps. Initially, there are no repressors or β-galacosidases in the system. Although the number of repressors increases over the first 200 iterations, it 3 Here the switch-off state is an emergent property, resulting from the interactions of multiple repressor proteins. LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.13

14 14 Jacob and Burleigh Figure 9. Concentrations of the biomolecular agents of the lactose simulation illustrated in Figure 8. cannot prevent the production of some β-galactosidase enzymes. However, once the repressor concentration has reached its initial peak, it completely blocks the lacz operator, which stops any further expression of β-galactosidase (around step 400). Shortly after iteration step 400, lactose is introduced into the cell, which almost immediately triggers the formation of repressor-lactose complexes. Now that free repressors are too few to block the operator, after a short delay the number of β-galacosidases increases, resulting in a rise of both glucose and galactose. The lifetime of lactose within the cell was set to 350 time steps, which reduces the lactose concentration to zero shortly before iteration step 800. This causes the repressor concentration to build up again and resume repressing β-galactosidase production, which brings the system back to its initial state with a high number of repressors and no β-galactosidase. LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.14

15 An Agent-based Gene Regulation Model Conclusion and Future Work We have presented a 3D agent-based model of the lac operon gene regulatory system, including a fast visualization engine. Currently, we work with both a Java3D and a C++/OpenGL version of our simulations. The model focuses on simulating important aspects of a biomolecular system including basic genetic processes such as transcription and translation. We believe that such simulations and visualizations will not only serve as powerful educational tools, but will also greatly support biologists in their understanding of complex gene regulatory systems, and decentralized, massively-parallel biological systems in general. A decentralized, swarm approach for modelling the lac operon closely approximates the way in which biologists view such systems. Although our simulations have so far only been tested for a relatively small number of (hundreds of) interacting agents, the system is designed to handle a much larger number of proteins and other cellular entities, thus getting closer to more accurate simulations of massivelyparallel interaction processes within a cell that involve hundreds of thousands of particles. The visualization, developed as a 2D projection on a normal computer screen, is further enhanced through stereoscopic 3D in a CAVE R immersive environment (Burleigh et al., 2003). On the other hand, we are also investigating how the number of biomolecular agents actually affects the emergent behaviour patterns, which we observe in our simulations and which can be measured in vivo in wet-lab experiments. Future work includes integrating additional aspects of the lac operon not covered in the current model. This includes the CAP Catabolite activator complex, that acts as an initiation factor for promoting the production of β-galactosidase based on glucose concentrations, and lactose permease, which facilitates the entry of lactose into the cell. Another important step is to tune the model towards experimental data derived from E. coli wetlab experiments. Our model also enables us to reconstruct other regulatory systems (such as the λ-switch (Ptashne and Gann, 2002) or the repressilator (Elowitz and Leibler, 2000)) and investigate general robustness properties of gene regulatory systems. We also plan to incorporate an evolutionary computation engine into the simulation, such as the Evolvica system (Jacob, 2001). Evolution of this gene system may lead to interesting and complex behaviours that can be compared with other gene regulatory systems evolved by nature. On the web site jacob/esd/lacoperon. LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.15

16 16 Jacob and Burleigh one will find further information about our lactose operon model and other swarm-based models of biological systems, such as the λ-switch and an artificial immune system. References Alberts, B., D. Bray, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter: 1998, Essential cell biology : an introduction to the molecular biology of the cell. New York: Garland. Allen, L. J. S.: 2003, An Introduction to Stochastic Processes with Applications to Biology. Upper Saddle River, NJ: Pearson Education. Beckwith, J. R. and D. Zipser (eds.): 1970, The Lactose Operon. Cold Spring Harbor, NY: Cold Spring Harbor Laboratory Press. Bonabeau, E., M. Dorigo, and G. Theraulaz: 1999, Swarm Intelligence: From Natural to Artificial Systems, Santa Fe Insitute Studies in the Sciences of Complexity. New York: Oxford University Press. Bower, J. M. and H. Bolouri (eds.): 2001, Computational Modeling of Genetic and Biochemical Networks. Cambridge, MA: MIT Press. Burleigh, I., G. Suen, and C. Jacob: 2003, DNA in Action! A 3D Swarm-based Model of a Gene Regulatory System. In: First Australian Conference on Artificial Life. Canberra, Australia. Collado-Vides, J.: 1992, Towards a grammatical paradigm for the study of the regulation of gene expression. In: B. Goodwin and P. Saunders (eds.): Theoretical Biology. Epigenetic and Evolutionary Order from Complex Systems. Baltimore, ML: Johns Hopkins University Press, pp Elowitz, M. B. and S. A. Leibler: 2000, Synthetic gene oscillatory network of transcriptional regulators. Nature 403, Jacob, C.: 2001, Illustrating Evolutionary Computation with Mathematica. San Francisco, CA: Morgan Kaufmann Publishers. Jacob, F. and J. Monod: 1961, Genetic regulatory mechanisms in the synthesis of proteins. Molecular Biology 3, Matsuno, H., A. Doi, A. Tanaka, H. Aoshima, Y. Hirata, and S. Miyano: 2001, Genomic Object Net: Basic Architecture for Representing and Simulating Biopathways. In: Ninth International Conference on Intelligent Systems for Molecular Biology. Copenhagen, Denmark. Müller-Hill, B.: 1996, The lac Operon - A Short History of a Genetic Paradigm. Berlin: Walter de Gryter. Ptashne, M. and A. Gann: 2002, Genes & Signals. Cold Spring Harbor, NY: Cold Spring Harbor Laboratory Press. Salzberg, S., D. Searls, and S. Kasif (eds.): 1998, Computational Methods in Molecular Biology, Vol. 32 of New Comprehensive Biochemistry. Amsterdam: Elsevier. Suen, G. and C. Jacob: 2003, A Symbolic and Graphical Gene Regulation Model of the lac Operon. In: Fifth International Mathematica Symposium. London, England, pp , Imperial College Press. Tomita, M., K. Hashimoto, K. Takahashi, Y. Matsuzaki, R. Matsushima, K. Saito, K. Yugi, F. Miyoshi, H. Nakano, S. Tanida, Y. Saito, A. Kawase, N. Watanabe, T. Shimizu, and Y. Nakayama: 2000, The E-CELL Project: Towards Integrative Simulation of Cellular Processes. New Generation Computing 18(1), LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.16

17 An Agent-based Gene Regulation Model 17 Watson, J. D. and F. H. C. Crick: 1953, A Structure for Deoxyribose Nucleic Acid. Nature 171, LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.17

18 LAC-Swarms-Final-Print.tex; 18/10/2004; 14:30; p.18