## Info

Number of docking points within threshold distance

Fig. 13. Average Time against na Fig. 14. Cumulative probability distri bution for the ligand-protein docking time

Fig. 13. Average Time against na Fig. 14. Cumulative probability distri bution for the ligand-protein docking time

Estimation of Tf. The next step is to estimate the mean of the time for rotation of the docking site of the ligand axis to produce the final docked complex. We obviously get T[ « 1 x 10"11 and 2 x 10"11 secs for the two wd estimates reported previously. Thus in general we can say that the time for rotation is too small in comparison to the time for collision, TC as reported subsequently. Thus the total time for ligand-protein docking is dominated by TC which corroborates the results reported in [19].

Dependence of T\ on At. Fig 12 plots T1 against different values for At. The average time for ligand-protein docking remains constant with increasing At. The same characteristics are seen for different number of docking points considered, ns = 8,15, 25 respectively. Though we have ns = 8 for the ligand-protein pair under consideration, we have reported the plots for different values of ns to show the dependence of the average binding time on ns . The activation energy, Eact is kept at 0 for the above plots. For, ns = 8, we find T1 = 0.000395 secs as against 0.00025 secs as estimated from the experimental rate constant value of 106M_1s_1. This is a very important finding from our model. It states that for the process of ligand-protein docking no activation energy is required, i.e. the ligand molecules do not have to overcome an energy barrier for successful docking. Indeed, biological experiments have indicated that the docking process occurs due to changes in monomer bonds into dimers and the resultant change in free energy is used for the rotational motion of the ligand to achieve the final docked conformation. Thus this finding corroborates the validity of our model. The results were generated assuming an average of 105 molecules of OMTKY inside the cell.

Also it can be noted that the average time for binding (= 0.000395 secs) is very high compared to our estimate of T[. Thus it can be inferred that the time taken for the rotational motion of the ligand is negligible in comparison to TC.

It is to be noted that pb as calculated above also corresponds to the number of collisions in time At of the ligand molecule with the protein. And for our assumption of at most one collision taking place in At to hold, we have to make sure that 0 < pb < 1 (this is also true because pb is a probability). Thus the regions to the right of the vertical lines corresponding to each ns plot denotes the forbidden region where pb > 1 even though 0 < p < 1. This gives us an estimate of the allowable At values for different ns's such that T1 indeed remains constant. Out estimates show that with At < 10~8, T1 remains constant for most values of ns.

Dependence of Ti on ns. Fig 13 plots T1 against the different possible ns values and we find that the average time for docking decreases as the total number of docking points ns is increased. This is again logical as the ligand molecules now have more options for binding resulting in a higher value of pf and subsequently pt.

The stochastic nature of the docking time. Fig 14 plots the cumulative distribution function (CDF) for the total time of binding with Eact = 0. The time for collision followed an exponential distribution (as the calculated mean was very close to the standard deviation). Also, because the T{ component is very small in comparison to TC, the overall time for binding can be approximated to follow an exponential distribution given by Eq 37. Note that incorporating T[ ^ TC in Eq 43 we get Eq 37 implying that the total time for docking is dominated by the exponential distribution outlined in Eq 37.

Fig 15 illustrates the dependence of the average time for docking (T1) on the number of protein (Human Leukocyte elastase) molecules in the cell for a fixed number of ligand (OMTKY) molecules (« 105). The corresponding time of reactions estimated from the experimental rate constant of 106Mhave also been reported. The docking time estimates from our theoretical model very closely matches the experimental estimates in the acceptable range of the number of protein molecules (varied from 103 — 109 molecules as can be found in any standard human cell).

### 4.3 Important Observations

1. Our model achieves the experimental rate constant estimate with zero activation energy requirement for the protein-ligand pair under consideration in human cells. The stochastic nature of protein-ligand binding time can be approximated by a general distribution with pdf of the form given in Eq 43 and first and second moments given by T1 and T2 respectively. However, for this protein-ligand pair, the total docking time can be approximated as an exponential distribution with pdf given by Eq 37 as T{ ^ Tf.

2. The average time for DNA-protein binding is independent of At and decreases as the length of the docking site increases (i.e., as ns increases).

3. An acceptable estimate of At is 10"8 secs. Fig 12 shows the dependence of the average time on At. We find that a wider range of At is available (keeping pb < 1) as ns decreases.

4. The mean of the total docking time (T1) decreases as the length of the docking site (ns) increases.

5. The average angle of rotation (0) for the ligand to reach the final docked conformation is very small. This coupled with the fact that the average angular velocity of the docking site on the ligand axis being very high makes the mean time taken for rotation negligible in comparison to the collision theory component of the docking time.

5 Discussion

### 5.1 Limitations of Our Model

Maxwell-Boltzman distribution of molecular velocities. The Maxwell-Boltzmann distribution gives a good estimate of molecular velocities and is widely used in practice. Molecular dynamic (MD) simulation measurements during protein reactions show that the velocity distribution of proteins in the cytoplasm closely match the Maxwell-Boltzmann distribution. However, its application in our collision theory model might not give perfect results. Ideally the velocity distribution should incorporate the properties of the cytoplasm, the pro-tein/ligand structure and also the electrostatic forces that come into play. We plan to extend our model to incorporate more realistic velocity distributions in the future.

3-D protein/ligand structure. Another point to note is that the pf estimation can be improved by considering the 3-D structures of the protein and the ligand. Ideally, the motifs of the protein/ligand molecules are located towards the outer surface such that our straight line assumption of the docking sites are quite realistic. However, the denominator in the expression for pf considers all possible atoms on the protein/ligand molecules. However, due to their 3-d structure, not all of these molecules are exposed towards the outer protein surface that the ligand can collide to. As a result our estimates of pf is actually a little lower than what should be a good estimate for the same, resulting in a corresponding decrease in pf and hence pt and a resultant increase in Tf and hence Ti. This might as well explain the slightly greater time reported from our model in comparison to the experimental estimates (recall that the experimental estimate was 0.00025 secs as against the 0.000395 secs reported by our model).

Straight line assumption of the docking sites on the protein/ligand backbones. As mentioned before, we have approximated the docking site on the protein/ligand axes as straight lines to simplify the computations of the average angle of rotation 0avg and subsequently the average time required for rotation, T{. However, because T{ ^ Tf, the T{ component of Ti is negligible and the results reported from our theoretical model are quite close to experimental estimates. We are working on this aspect to identify a better estimate of T{ that models the actual docking process more closely.

### 5.2 Biological Implications

Several ligands coming into the cell for docking. If we consider several lig-ands searching for their docking sites on the protein simultaneously, our results still remain valid. Note that as the number of ligands increase in the cell, the binding rate will increase. Assuming the docking time to be completely characterized by the collision theory part, an analytical estimate of the binding rate in such cases can be achieved by using the batch model of [11]. However, the time taken for any particular ligand to bind to the corresponding protein molecule still remains the same. Thus increasing the number of ligands should not change the results that we report for any particular ligand. In fact, this discrepancy arises because of the definition of the binding rate the inverse of which gives the time required for a successful docking to occur between the protein-ligand pair. Looking into the problem from one specific ligand's perspective (as we do in this paper), the average time required for docking will be the same assuming there are enough number of protein molecules in the cell. This is a salient feature of our stochastic simulation paradigm where we track the course of events initiated by any particular molecule in the cell to study the dynamics of the entire cell. However, this may cause molecular crowding (of ligands) in the cell which can have an impact on the search time. Further studies are required to cover this aspect of ligand-protein docking.

Funnels and local organization of sites. Local arrangement of the binding sites of proteins tend to create a funnel in the binding energy landscape leading to more rapid binding of cognate sites. Our model assumes no such funnels of energy field. If the ligands spend most of their search time far from the cognate site our model will remain valid and no significant decrease in binding time is expected.

### 6 Conclusion

We have presented a computationally simplified model to estimate the ligand-protein binding time based on collision theory. The motivation for this simplified model is to construct a simulation that can model a complex biological system which is currently beyond the scope of kinetic rate based simulations. The model is robust enough as the major contributing factors (molecular motion) are captured in a reasonably accurate way for general cell environments. For an extreme cell environment condition, where the influence of the electrostatic force will be significantly different, the model will not provide such accuracy. We are exploring the possibility to modify the velocity distribution to capture the effect of this extreme cell environment. However, the model is computationally fast and allows our stochastic simulator to model complex biological systems at the molecular level (i.e., that involves many such docking events). The proposed mechanism is not only limited to protein-ligand interactions but provide a general framework for protein-DNA binding. The complexity of the 3-d protein/ligand structures have been simplified in this paper to achieve acceptable estimates of the holding time of the ligand-protein binding event. We found that no activation energy is required for the docking process and the rotational energy for ligand-protein complex to attain the final docked conformation is contributed by the total change in free energy of the complex. The proposed mechanism has important biological implications in explaining how a ligand can find its docking site on the protein, in vivo, in the presence of other proteins and by a simultaneous search of several ligands. Besides providing a quantitative framework for analysis of the kinetics of ligand-protein binding, our model also links molecular properties of the ligand/protein and the structure of the docking sites on the ligand/protein backbones to the timing of the docking event. This provides us with a general parametric model for this biological function for our discrete-event based simulation framework. Once the model is validated for a few test cases, it can serve as a parametric model that can be used for all ligand-protein binding scenarios where the binding details are available. This may eliminate the necessity of conducting specific experiments for determining the rate constants to model a complex biological process.

References

1. Human Genome Project,

http://www.ornl.gov/sci/techresources/Human_Genome/home.shtml

2. Schena, M.: Microarray Analysis, ISBN: 0471414433 (2002)

3. Duggan, D.J., Bittner, M., Chen, Y., Meltzer, P., Trent, J.M.: Expression profiling using cDNA microarrays. Nature Genetics Supplement 21, 10-14 (1999)

4. McCulloch, A.D., Huber, G.: Integrative biological modeling in silico. In Silico Simulation of Biological Processes, Novartis Foundation Symposium 247 (2002)

5. Bower, J.A., Bolouri, H.: Computational Modeling of Genetic and Biological Network. MIT Press, Cambridge (2001)

6. Hunter, P., Nielsen, P., Bullivant, D.: In Silico Simulation of Biological Processes. In: Novartis Foundation Symposium No. 247, pp. 207-221. Wiley, Chichester (2002)

7. The RCSB Protein Data Bank, http://www.rcsb.org/pdb/

8. von Hippel, P.H., Berg, O.G.: On the specificity of DNA-protein interactions. In: Proc. Natl. Acad. Sci., USA, vol. 83, pp. 1608-1612 (1986)

9. Ghosh, S., Ghosh, P., Basu, K., Das, S., Daefler, S.: SimBioSys: A Discrete Event Simulation Platform for 'in silico' Study of Biological Systems. In: Proceedings of 39th IEEE Annual Simulation Symposium, Huntsville, AL (April 2 - 6, 2006)

10. Ghosh, P., Ghosh, S., Basu, K., Das, S., Daefler, S.: An Analytical Model to Estimate the time taken for Cytoplasmic Reactions for Stochastic Simulation of Complex Biological Systems. In: 2nd IEEE Granular Computing Conf., USA (2006)

11. Ghosh, P., Ghosh, S., Basu, K., Das, S., Daefler, S.: Stochastoc Modeling of Cyto-plasmic Reactions for Complex Biological Systems. In: IEE International Conference on Computational Science and its Applications, Glasgow, Scotland, May 8-11 (2006)

12. Ghosh, S., Ghosh, P., Basu, K., Das, S.K.: iSimBioSys: An 'In Silico' Discrete Event Simulation Framework for Modeling Biological Systems. IEEE Comp. Systems BioInf. Conf. (2005)

13. Hasty, J., Collins, J.J.: Translating the Noise. Nature 31, 13-14 (2002)

14. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25), 2340-2361 (1977)

15. Kitano, H.: Cell Designer: A modeling tool of biochemical networks. online at, http://www.celldesigner.org/

16. Adalsteinsson, D., McMillen, D., Elston, T.C.: Biochemical Network Stochastic Simulator (BioNets): software for stochastic modeling of biochemical networks. BMC Bioinformatics (March 2004)

17. Le Novre, N., Shimizu, T.S.: StochSim: modeling of stochastic biomolecular processes. Bioinformatics 17, 575-576 (2000)

18. Cell Illustrator. online at, http://www.fqspl.com.pl/

19. Camacho, C.J., Kimura, S.R., DeLisi, C., Vajda, S.: Kinetics of Desolvation-Mediated Protein-Protein Binding. Biophysical Journal 78, 1094-1105 (2000)

20. DeLisi, C., Wiegel, F.: Effect of nonspecific forces and finite receptor number on rate constants of ligand-cell-bound-receptor interactions. In: Proc. Natl. Acad. Sci., 78th edn., USA, pp. 5569-5572 (1981)

21. Smoluchowski, M.V.: Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Loeschungen. Z. Phys. Chem. 92, 129-168

22. Northrup, S.H., Erickson, H.P.: Kinetics of proteinprotein association explained by Brownian dynamics computer simulations. In: Proc. Natl. Acad. Sci., USA, vol. 89, pp. 3338-3342 (1992)

23. Fischer, H., Polikarpov, I., Craievich, A.F.: Average protein density is a molecular-weight-dependent function. Protein Science 13, 2825-2828 (2004)

24. Sobolev, V., Sorokine, A., Prilusky, J., Abola, E.E., Edelman, M.: Automated analysis of interatomic contacts in proteins. Bioinformatics 15, 327-332 (1999)

25. Nanomedicine, vol. I: Basic Capabilities, http://www.nanomedicine.com/NMI/ 3.2.5.htm

26. Camacho, C., Weng, Z., Vajda, S., De Lisi, C.: Biophisics J, vol. 76, pp. 1166-1178 (1999)

27. Camacho, C., De Lisi, C., Vajda, S.: Thermodynamics of the Drug-Receptor Interactions. In: Raffa, R. (ed.) Wiley, London (2001)

28. Camacho, C., Vajda, S.: Protein docking along smooth association pathways. PNAS 98(19), 10636-10641 (2001)

29. Sharp, K., Fine, R., Honig, B.: Computer simulations of the diffusion of a substrate to an active site of an enzyme. Science 236, 1460-1463 (1987)

30. Stone, R., Dennis, S., Hofsteenge, J.: Quantitative evaluation of the contribution of ionic interactions to the formation of thrombin-hirudin complex. Biochemistry 28, 6857-6863 (1989)

31. Eltis, L., Herbert, R., Barker, P., Mauk, A., Northrup, S.: Reduction of horse ferricytochrome c by bovine liver ferrocytochrome b5. Experimental and theoretical analysis. Biochemistry 30, 3663-3674 (1991)

32. Schreiber, G., Fersht, A.: Rapid, electrostatically assisted association of proteins. Nature Struct. Biol. 3, 427-431 (1996)

33. Gabdoulline, R., Wade, R.: Simulation of the diffusional association of barnase and barstar. Biophisics J. 72, 1917-1929 (1997)

34. Vijaykumar, M., Wong, K., Schreiber, G., Fersht, A., Szabo, A., Zhou, H.: Electrostatic enhancement of diffusion-controlled protein-protein association: comparison of theory and experiment on Barnase and Barstar. J. Mol. Biol. 278, 1015-1024 (1998)

35. Chothia, C., Janin, J.: Principles of protein-protein recognition. Nature 256, 705708 (1975)

36. Camacho, C., Weng, Z., Vajda, S., DeLisi, C.: Free energy landscapes of encounter complexes in protein-protein association. Biophisics J. 76, 1166-1178 (1999)

37. Tomita, M., et al.: ECell: Software environment for whole cell simulation. Bioin-formatics 15(1), 72-84 (1999)

38. Sauro, H.M.: Jarnac: a system for interactive metabolic analysis. Animating the Cellular Map. In: 9th International BioThermoKinetics Meeting, Stellenbosch University Press, ch. 33, pp. 221-228 (2000)

39. Ghosh, P., Ghosh, S., Basu, K., Das, S., Daefler, S.: Modeling the Diffusion process in Stochastic Event based Simulation of the PhoPQ system. International Symposium on Computational Biology and Bioinformatics (ISBB), India (December 2006)

40. Ghosh, P., Ghosh, S., Basu, K., Das, S.K.: Modeling protein-DNA binding time in Stochastic Discrete Event Simulation of Biological Processes. submitted to the Re-comb Satellite Conference on Systems Biology, San Diego, USA (November 2006)

41. Regev, A., Silverman, W., Shapiro, E.: Representation and simulation of biochemical processes using the ^-calculus process algebra. In: Proceedings of the Pacific Symposium of Biocomputing 2001 (PSB2001), vol. 6, pp. 459-470

42. Priami, C., Regev, A., Silverman, W., Shapiro, E.: Application of a stochastic name passing calculus to representation and simulation of molecular processes. Information Processing Letters 80, 25-31

43. Regev, A.: Representation and simulation of molecular pathways in the stochastic ^-calculus. In: Proceedings of the 2nd workshop on Computation of Biochemical Pathways and Genetic Networks (2001)

44. Regev, A., Silverman, W., Shapiro, E.: Representing biomolecular processes with computer process algebra: ^-calculus programs of signal transduction pathways. In: Proceedings of the Pacific Symposium of Biocomputing 2000, World Scientific Press, Singapore

45. Finn, R.D., Marshall, M., Bateman, A.: iPfam: visualization of protein-protein interactions in PDB at domain and amino acid resolutions. Bioinformatics 21, 410-412 (2005)

46. Fogler, H., Gurmen, M.: Elements of Chemical Reaction Engineering. ch. 3.1, online at http://www.engin.umich.edu/cre/03chap/html/collision/

Equation-Based Congestion Control in the Internet Biologic Environment

Morteza Analoui and Shahram Jamali

IUST University, Narmak, Tehran, Iran {analoui, jamali}@iust.ac.ir

Abstract. In this paper we present a new aspect of human kind life that is the Internet. We call this new ecosystem IBE (Internet Biologic Environment). As a first step in modeling of IBE we view it from point of congestion control and develop an algorithm that utilizes some aspects of biologically inspired mathematical models as a nontraditional approach to design of congestion control in communication networks. We show that the interaction of those Internet entities that involved in congestion control mechanisms is similar to predator-prey and competition interactions. We combine the mathematical models of predator-prey and competition interactions to obtain a hybrid model and apply it in congestion control issue. Simulation results show that using appropriately defined parameters, this model leads to a stable, fair and highperformance congestion control algorithm.

Keywords: Communication Networks, Congestion Control, Bio-Inspired Computing, Competition and Predator-Prey.

### 1 Introduction

The human kind is confronting a new biological life, which is the Internet. The Internet as a new environment has a variety of species. Internet species include the human, applications, software, computers, protocols, algorithms and so on. The interactions of these species determine the dynamics of this ecology. We believe that the consideration of the Internet as a biologic environment can originate two areas of benefits:

### Bio-Inspired Network Control

Technology is taking us to a world where myriads of heavily networked devices interact with the physical world in multiple ways, and at multiple scales, from the global Internet scale down to micro- and nano-devices. A fundamental research challenge is the design of robust decentralized computing systems capable of operating under changing environments and noisy input, and yet exhibits the desired behavior and response time. These systems should be able to adapt and learn how to react to unforeseen scenarios as well as to display properties comparable to social entities. Biological systems are able to handle many of these challenges with an

C. Priami (Ed.): Trans. on Comput. Syst. Biol. VIII, LNBI 4780, pp. 42 - 62, 2007. © Springer-Verlag Berlin Heidelberg 2007

elegance and efficiency still far beyond current human artifacts. Based on this observation, bio-inspired approaches have been proposed in the past years as a strategy to handle the complexity of such systems.

Network-Based Biology Analysis, Design and Control

Network analysis and modeling address a wide spectrum of techniques for studying artificial and natural networks. It refers domains consisting of individuals that are linked together into complex networks, communication networks, social networks and biological networks. They constitute a very active area of research in a variety of scientific disciplines, including communication, Biology, Artificial Intelligence and Mathematics. More recently, the study of communication networks has gained increased attention in modeling diverse areas of Internet, such as performance, security, congestion control, and so on. We believe that the techniques developed for the analysis of Internet can provide a substantial background for studying the structure, dynamics and evolution of complex biological networks. Hence, the biologists can borrow this background to study aspects such as population dynamics, fault tolerance, adaptability, complexity, information flow, community structures, and propagation patterns.

In order to study and analysis Internet Biologic Environment (IBE), we need to model it. The following models can be proposed for IBE: Performance Model, Management Model, Security Model, Congestion Control Model and so on. In this paper we are going to model this new biological life of the human kind. As the first step the congestion control modeling is considered. The central aim of this paper is to obtain bio-inspired methods to engineer congestion control.

Previous Internet research has been heavily based on measurements and simulations, which have intrinsic limitations. For example, network measurements cannot tell us the effects of new protocols before their deployment. Simulations only work for small networks with simple topology due to the constraints of the memory size and processor speed. We cannot assume that a protocol that works in a small network will still perform well in the Internet [4]. A theoretical framework and especially mathematical models can greatly help us understand the advantages and shortcomings of current Internet technologies and guide us to design new protocols for identified problems and future networks. The steady-state throughput of TCP Reno has been studied based on the stationary distribution of congestion windows, e.g. [5, 6, 7, 8]. These studies show that the TCP throughput is inversely proportional to end-to-end delay and to the square root of packet loss probability. Padhye [9] refined the model to capture the fast retransmit mechanism and the time-out effect, and achieved a more accurate formula. This equilibrium property of TCP Reno is used to define the notion of TCP-friendliness and motivates the equation based congestion control [10]. Misra [11, 12] proposed an ordinary differential equation model of the dynamics of TCP Reno, which is derived by studying congestion window size with a stochastic differential equation. This deterministic model treats the rate as fluid quantities (by assuming that the packet is infinitely small) and ignores the randomness in packet level, in contrast to the classical queuing theory approach, which relies on stochastic models. This model has been quickly combined with feedback control theory to study the dynamics of TCP systems, e.g. [13, 14], and to design stable AQM (Active Queue Management) algorithms, e.g. [15, 16, 17, 18, 19]. Similar flow models for other TCP schemes are also developed, e.g. [20, 21] for TCP Vegas, and [22, 23] for FAST TCP. The analysis and design of protocols for large-scale network have been made possible with the optimization framework and the duality model. Kelly [24, 25] formulated the bandwidth allocation problem as a utility maximization over source rates with capacity constraints. A distributed algorithm is also provided by Kelly [25] to globally solve the penalty function form of this optimization problem. This algorithm is called the primal algorithm where the sources adapt their rates dynamically, and the link prices are calculated by a static function of arrival rates. Low and Lapsley [26] proposed a gradient projection algorithm to solve its dual problem. It is shown that this algorithm globally converges to the exact solution of the original optimization problem since there is no duality gap. This approach is called the dual algorithm, where links adapt their prices dynamically, and the users' source rates are determined by a static function.

In the current paper, we use some well-established biological mathematical models, and apply them to congestion control scheme of communication networks to gain a bio-inspired equation-based congestion control algorithm.

In section 2, we present a conceptual framework for Biologically Inspired network Control (BICC) and a literature about the models of interacting populations and explain analogy between the biological interaction and the communication networks. Section 3 presents a case study and introduces a methodology for applying the predator-prey to the Internet congestion control scheme. It gives an example and discusses the stability, the fairness and the performance of the introduced solution for its congestion control algorithm. In Section 4 we present another case study for applying the combinational model of competition and predator-prey models to the Internet congestion control scheme. The implementation consideration for the proposed algorithms will be given in section 5. In section 6 we talk about "how the system biology benefits from this work". We conclude in section 7 with future works.

2 A Conceptual Framework: Internet as an Ecosystem Analogy

The first step in bio-inspired computation should be to develop more sophisticated biological models as sources of computational inspiration, and to use a conceptual framework to develop and analyze the computational metaphors and algorithms. We believe that bio-inspired algorithms are best developed and analyzed in the context of a multidisciplinary conceptual framework that provides for sophisticated biological models and well-founded analytical principles. In the reverse direction, the techniques, algorithms, protocols and analytical models, etc. that are developed for analysis of Internet, provide a significant background for research in bionetwork area.

Fig. 1 illustrates a possible structure for such a conceptual framework. Here probes (observations and experiments) are used to provide a (partial and noisy) view of the complex biological system. From this limited view, we build and validate simplifying abstract representations and models of the biology. From these biological models, we build and validate analytical computational frameworks. Validation may use mathematical analysis, benchmark problems, and engineering demonstrators. These frameworks provide principles for designing and analyzing bio-inspired algorithms

applicable to non-biological problems, possibly tailored to a range of problem domains and contain as much or as little biological realism as appropriate. The concept flow also supports the design of algorithms specifically tailored to modeling the original biological domain, permits influencing and validating the structure of the biological models, and can help suggest ideas of further experiments to probe the biological system. This is necessarily an interdisciplinary process, requiring collaboration between (at least) biologists, mathematicians, and computer scientists to build a complete framework.

### 2.1 Internet Ecology

Consider a network with a set of k source nodes and a set of k destination nodes. We denote S={Sj, S2, Sk} as the set of source nodes with identical round-trip propagation delay (RTT), and D={D1, D2, ..., Dk} as the set of destination nodes. Our network model consists of a bottleneck link from LAN to WAN as shown in Fig. 2 and uses a window-based algorithm for congestion control. The bottleneck link has capacity of B packet per RRT. The congestion window (W) is a sender-side limit on

Ecosystem: The Internet |

## Post a comment