Areejit Samal Regulation

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Areejit Samal Regulation

  1. 1. System level dynamics and robustness of the genetic network regulating E. coli metabolism Areejit Samal Department of Physics and Astrophysics University of Delhi Delhi 110007 India
  2. 2. Outline• Background• System: E. coli transcriptional regulatory network controlling metabolism (iMC1010v1)• Simulation results• Design features of the regulatory network• Conclusions June 15, 2009 Areejit Samal
  3. 3. Cell Gene A5’ 3’ Promoter Coding region Gene B 5’ 3’DNA Promoter Coding region Gene C mRNA 5’ 3’ Promoter Coding region mRNA Transcriptional Regulatory mRNA NetworkProtein Protein Protein A Protein B Protein C Interaction Network Metabolic NetworkMetabolite A B C D Metabolic Pathway Cell can be viewed as a ‘network of networks’
  4. 4. CellEnvironment Gene A 5’ 3’ Promoter Coding region Gene B 5’ 3’ DNA Promoter Coding region Gene C mRNA 5’ 3’ Promoter Coding region mRNA Transcriptional Regulatory mRNA Network Protein Protein Protein A Protein B Protein C Interaction Network Metabolic Network Metabolite A B C D Metabolic Pathway Cell can be viewed as a ‘network of networks’
  5. 5. Boolean network approach to model Gene Regulatory Networks• Boolean networks were introduced by Stuart Kauffman as a framework to study dynamics of Genetic networks.• In this approach, gene expression is quantized to two levels: – on or active (represented by 1) and – off or inactive (represented by 0).• Each gene at any point of time is in one of the two states (i.e. active or inactive).• In this approach, time is taken as discrete.• Also, the expression state of each gene at any time instant is determined by the state of its input genes at the previous time instant via a logical rule or update function.
  6. 6. Simplified Diagram of the Transcriptional Regulatory Network controlling metabolism • An input may activate or repress the expression of the gene. For example: Gene B [t+1] = NOT Gene A [t] • When there are more than one input to a gene, the expression state of the gene will be determined by the state of the inputs based on a logical rule. • This logical rule may be expressed in terms of Boolean operators (AND, OR, NOT). • For example: Gene C [t+1] = Gene A [t] AND NOT Gene B [t] • The state of Gene C determines if the metabolic reaction can occur inside the cell. Metabolic reaction June 15, 2009 Areejit Samal
  7. 7. Modelling Gene Regulatory Networks as Random Boolean Networks In the absence of data on real genetic networks, Boolean networks have been used primarily to study the dynamics of the genetic networks that were – either members of ensemble of random networks or – networks generated using the knowledge of the connectivity of genes and TF in an organism along with random Boolean rules at each node as input function governing the output state of the gene June 15, 2009 Areejit Samal
  8. 8. E. coli transcriptional regulatory network controlling metabolism (iMC1010v1) In this work, we have studied the database iMC1010v1 containing the transcriptional regulatory network (TRN) controlling E. coli metabolism has become available. The network contained in the database was reconstructed from primary literature sources. The database iMC1010v1 contains the following types of information: – the connections between genes and transcription factors (TF) – dependence of genes and TF activity based on presence or absence of external metabolites or nutrients in the environment – the Boolean rule describing the regulation of each gene as a function of the state of the input nodes Available at: Bernhard Palsson’s Group Webpage (http://gcrg.ucsd.edu/) June 15, 2009 Areejit Samal
  9. 9. Schematic of Transcriptional Regulatory  Network controlling metabolism5’ Gene A 3’ Promoter Coding region 5’ Gene B 3’DNA Promoter Coding region mRNA 5’ Gene C 3’ Promoter Coding region mRNA Transcriptional Regulatory mRNA NetworkProtein Protein A Protein B Protein C Metabolic Network C D June 15, 2009 Metabolic Areejit Samal reaction
  10. 10. Description of the E. coli TRN controlling metabolism (iMC1010v1)• There are 583 genes in this network which can be further subdivided into – 479 genes that code for metabolic enzymes – 104 genes that code for TF• The state of these 583 genes is dependent upon – the state of 103 TF and – presence or absence of 96 external metabolites• The database provides a Boolean rule for each of the 583 genes contained in the network. June 15, 2009 Areejit Samal
  11. 11. The pink nodes  represent genes  coding for TF, brown  nodes represent  genes that code for  metabolic enzymes  and the green nodes  represent external  metabolites.  The complete  network can be  subdivided into a  large connected  component and few  small disconnected  components.June 15, 2009 Areejit Samal
  12. 12. Example of an input function in form of a Boolean rule controlling the output state of a gene A B C Truth Table b2731 o2(e) b3202 A B C OUTPUT 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 b2720 1 0 1 0 1 1 0 1 OUTPUT 1 1 1 0 b2720[t+1] = IF ( b2731[t] AND b3202[t] AND NOT o2(e)[t]) June 15, 2009 Areejit Samal
  13. 13. The Dynamical SystemWe have used the information in the database to construct the followingdiscrete dynamical system:   gi (t  1)  Gi ( g (t ), m)i  1...583gi (t  1) denotes the state of ith gene at time t+1 that is either 1 or 0.g (t ) is vector that collectively denotes the state of all genes at time tm is a vector of 96 elements (each 0 or 1) determining the state of the environmentGi contains all the information regarding the internal wiring of the network as well as the regulatory logic Areejit Samal June 15, 2009
  14. 14. State of the genetic networkThe state of the 583 genes at any given time instant gives the state of thenetwork. g(t)  g1 (t )   g (t )  where gi(t) = 0 or 1; i = 1 …. 583  2  Since each gene at any given time instant can be in one  g3 (t )  of the two states (0 or 1), the size of the state space is   .  2583. .    .   g (t )   583    June 15, 2009 Areejit Samal
  15. 15. State of the environmentThe presence or absence of the 96 external metabolites decide the state of theenvironment. m where mi = 0 or 1; i = 1 …. 96  m1  If an external metabolite or nutrient is present in the external m  environment, then we set the mi corresponding to it equal to 1  2  or else 0. .    In general, the concentration of external metabolites change .  with time.  m96  In the present study, we have considered buffered minimal   media (i.e., vector m constant in time). June 15, 2009 Areejit Samal
  16. 16. E. coli TRN controlling metabolism as a Boolean dynamical systemStuart Kauffman (1969,1993) studied dynamical systems of the form:  gi (t  1)  Gi ( g (t ))
  17. 17. E. coli TRN controlling metabolism as a Boolean dynamical systemStuart Kauffman (1969,1993) studied dynamical systems of the form:  gi (t  1)  Gi ( g (t ))The present database allowed us to systematically account for the effect of presence orabsence of nutrients in the environment on the dynamics of the regulatory network.   g i (t  1)  Gi ( g (t ), m )
  18. 18. Attractors of the E. coli TRN• In the Boolean approach, the configuration space of the system is finite. The discrete deterministic dynamics ensures that the system eventually returns to a configuration which it had at a previous time instant. The sequence of states that repeat themselves periodically is called an attractor of the system.• Starting from any one of the 2583 vectors as the initial configuration of genes and a fixed environment, the system can flow to different attractors for different initial configuration of genes. June 15, 2009 Areejit Samal
  19. 19. The Network exhibits stability against perturbations of gene configurations for a fixed environment   gi (t  1)  Gi ( g (t ), m) Start with different g(t) as initial configuration of Fix m to some buffered genes, and determine the attractor for the system minimal media e.g. Glucose for each initial configuration of genes. aerobic condition Question 1: How many attractors of the system do we obtain starting from different initial configuration of genes and for a fixed environment? Answer 1: We found that the attractors of the genetic network were typically fixed points or two cycles. For a given environment, the number of different attractors were up to 8 fixed points and 28 two cycles. However, the maximum hamming distance between any two attractor states for a given environment was 21. Hence, the states of most genes (≥562) was same in all attractor states for a given environment. We found that the network exhibits homeostasis or stability against perturbations of initial gene configurations for a fixed environment. June 15, 2009 Areejit Samal
  20. 20. Cellular Homeostasis 600 The graph shows that starting Random initial condition from even a initial Hamming inverse of the attractor configuration of genes that isHamming distance w.r.t. glucose 500 Attractor for glutamate aerobic medium aerobic condition attractor Attractor for acetate aerobic medium inverse of the attractor for the 400 glucose aerobic minimal media the system reaches the 300 attractor in four time steps. 200 Thus, any perturbation of gene configurations will be 100 washed out in few time steps and the system is robust to 0 such perturbations. 0 1 2 3 4 Time June 15, 2009 Areejit Samal
  21. 21. E. coli TRN exhibits flexibility of response under changing environmental conditions   gi (t  1)  Gi ( g (t ), m)Determine the attractors of the genetic system for Vary m across a set of 15427different environments m buffered minimal mediaQuestion 2: How different are the attractors from each other for various environmentalconditions?Answer 2: We obtained the attractors of the system starting with 15,427 environmentalconditions. The largest hamming distance obtained between two attractors corresponding todifferent environmental conditions was 145.The system shows flexibility of response to changing environmental conditions.We found that the system is insensitive to fluctuations in gene configurations for a given fixedexternal environment while it can shift to a different attractor when it encounters a change inthe environment. These properties ensure a robust dynamics of the underlying network. June 15, 2009 Areejit Samal
  22. 22. Flexibility of response 3x106 3x106 The graph shows that  2x106 the largest hamming  distance between two Frequency 2x106 136 138 140 142 144 146 attractors from a set of  attractors for 15,427  1x106 environmental  conditions was 145. 500x103 0 0 20 40 60 80 100 120 140 Hamming distance June 15, 2009 Areejit Samal
  23. 23. Flexibility of response 250 200 Each gene takes a value 0 or 1 in  the 15427 attractors for the Number of Genes different environmental  150 conditions. The standard  deviation of a gene’s value  100 across 15427 attractors is a  measure of the gene’s variability  50 across environmental conditions. 0 0 0 - 0.1 0.1 - 0.2 0.2 - 0.3 0.3 - 0.4 0.4 - 0.5 Standard deviation June 15, 2009 Areejit Samal
  24. 24. Functional significance of attractors of TRN controlling metabolism 1  Gene 1 is active: The enzyme is present to carry out a reaction in the metabolic network 0 Metabolic enzymes Gene 2 is inactive: The enzyme is absent and a reaction cannot happen in the network   1    0  The attractor of the genetic network for a given  .  environment constrains the set of active enzymes that    catalyze various reactions in the metabolic network .  .  TF   1    Attractor for a given environment June 15, 2009 Areejit Samal
  25. 25. Flux Balance Analysis (FBA) INPUT OUTPUTList of metabolic reactions withstoichiometric coefficients Growth rate for the Flux Balance given mediumBiomass composition Analysis (FBA) Fluxes of all reactionsMedium of growth orenvironment Reference: Varma and Palsson, Biotechnology (1994) June 15, 2009 Areejit Samal
  26. 26. Incorporating regulatory constraints within FBA INPUT OUTPUT Growth rate (pure) List of metabolic reactions Flux Balance AnalysisBiomass composition (FBA) Fluxes of all reactionsMedium of growth orenvironment June 15, 2009 Areejit Samal
  27. 27. Incorporating regulatory constraints within FBA INPUT OUTPUT Growth rate (pure) List of metabolic reactions Flux Balance AnalysisBiomass composition (FBA) Fluxes of all reactionsMedium of growth orenvironment m State of the environment June 15, 2009 Areejit Samal
  28. 28. Incorporating regulatory constraints within FBA INPUT OUTPUT Growth rate (pure) List of metabolic reactions Flux Balance AnalysisBiomass composition (FBA) Fluxes of all reactionsMedium of growth orenvironment 1  0   1    m 0 .    State of the .  environment .    1    Attractor of the genetic network June 15, 2009 Areejit Samal
  29. 29. Incorporating regulatory constraints within FBA INPUT OUTPUT Subset Growth rate (pure) List of metabolic reactions Flux Balance AnalysisBiomass composition (FBA) Fluxes of all reactionsMedium of growth orenvironment 1  0   1    m 0 .    State of the .  environment .    1    Attractor of the genetic network June 15, 2009 Areejit Samal
  30. 30. Incorporating regulatory constraints within FBA INPUT OUTPUT Subset Growth rate (pure) List of metabolic reactions Flux Balance Growth rate (constrained) AnalysisBiomass composition (FBA) Fluxes of all reactionsMedium of growth orenvironment 1  0   1    The ratio of constrained FBA growth m 0 .  rate to pure FBA growth rate is ≤ 1.   State of the .  environment .    1    Attractor of the genetic network June 15, 2009 Areejit Samal
  31. 31. Adaptability Question 3(a): What is the ratio of the constrained FBA growth rate to pure FBA growth rate for various environmental conditions? In other words, is the regulatory network reaching an attractor that can make optimal use of the underlying metabolic network? 7000 6000 Answer 3(a): Histogram of the ratio of constrained FBA growth rate in the attractor of 5000Number of media 4000 each of 15427 minimal media 3000 to the pure FBA growth rate 2000 in that medium. This is peaked at the bin with the 1000 largest ratio ≥ 0.9. 0 0 - 0.1 0.1 - 0.2 0.2 - 0.3 0.3 - 0.4 0.4 -0.5 0.5 - 0.6 0.6 - 0.7 0.7 - 0.8 0.8 - 0.9 0.9 -1.0 Ratio of constrained FBA growth rate to pure FBA growth rate June 15, 2009 Areejit Samal
  32. 32. Adaptability 1  1  . 1  1  0  . 1         0  1  . 0         1  0  . 0  .  .  . .         .  .  . .  .  .  . .         0    1    .  1    t=0 t=1 t=∞  m FBABiomasscomposition GR(t=0) GR(t=1) GR(t=∞) June 15, 2009 Areejit Samal
  33. 33. Adaptability 1  1  . 1  1  0  . 1  Question 3 (b): How well is the attractor of any particular        medium “adapted” to that medium? Does the movement to the 0  1  . 0         attractor “improve” the cell’s “metabolic functioning” in the 1  0  . 0  .  .  . .  medium?        .  .  . .  1.4 .  .  . .  Glutamine aerobic medium Answer 3(b):        1.2 Lactate aerobic medium Growth rate 0    1    .  1    Fucose aerobic medium Acetate aerobic medium 1.0 increases by a factor t=0 t=1 t=∞ Growth rate 0.8 of 3.5, averaged over pairs of minimal 0.6 media 0.4 From one minimal medium to another  0.2 the average time m FBA 0.0 taken to reach theBiomass 0 1 2 3 4 5 attractor is only 2.6composition Time steps Thus the regulatory dynamics enables the cell to adapt to its environment to improve its metabolic efficiency very GR(t=0) GR(t=1) GR(t=∞) substantially, fairly quickly. June 15, 2009 Areejit Samal
  34. 34. The graph shows  the genetic  network  controlling E. coli metabolism.June 15, 2009 Areejit Samal
  35. 35. Design Features of the network explain Homeostasis and Flexibility External Metabolites Transcription factors Metabolic GenesJune 15, 2009 Areejit Samal
  36. 36. Design Features of the network explain Homeostasis and Flexibility External MetabolitesThis is an acyclic graph with maximal depth 4. Fixing the environment leads to fixing of TF states and also the leaf nodes leading to homeostasis. But when  Transcription factorswe change the environment, then the attractor state changes endowing system with the property of flexible  Metabolic Genesresponse. June 15, 2009 Areejit Samal
  37. 37. Design Features of the network explain Homeostasis and Flexibility External MetabolitesThe very few feedbacks  Internal Metabolitesfrom metabolism on to transcription factors  are through the concentration of internal metabolites. Transcription factors Metabolic Genes June 15, 2009 Areejit Samal
  38. 38. Modularity, Flexibility and Evolvability This is a highly disconnected structure. The disconnected components are dynamically independent and hence can be regarded as modules. Such a structure can facilitate during evolution to new environmental niches.June 15, 2009 Areejit Samal
  39. 39. Almost all input functions in the E. coli TRN are canalyzing functions• When a gene has K inputs, then in general there can be 2 to the power of 2K input Boolean functions that can exist. – As K increases the number of possible Boolean functions also increases.• A Canalyzing Boolean function has at least one input such that for at least one input value for that input the output value is fixed.• Stuart Kauffman proposed that Canalyzing Boolean functions are likely to be over-represented in the real networks.• We found that all except four Boolean functions in the E. coli TRN were canalyzing. June 15, 2009 Areejit Samal
  40. 40. Design Features of the network• The genetic network regulating E. coli metabolism is – Largely acyclic – Hierarchical – Root control with environmental variables – Disconnected and modular structure at the level of transcription factors – Preponderance of canalyzing Boolean functions• There are some small cycles that exist due of presence of control by fluxes or internal metabolites but these cycles are very localized.• Note that cycles are expected in developmental systems such as cell cycle which is a temporal phenomena.• In metabolism, lack of cycles at the genetic level can be an advantage as this is a slow process.• Most cycles in metabolism exist at the level of enzymes and internal metabolites such a process is faster. June 15, 2009 Areejit Samal
  41. 41. Dynamics of the E. coli TRN controlling metabolism is highly ordered in contrast to that of Random Boolean Networks Reference: S.A. Kauffman (1993)Kauffman found that Random Boolean Networks (RBN) with K=2 are at the edge of chaos using Derrida Plot. Derrida plot is the discrete analog of the Lyapunovcoefficient. Derrida plot for RBNs with K>2 are found to be above the diagonal and their dynamics is quite chaotic.  June 15, 2009 Areejit Samal
  42. 42. Derrida Plot 1  0  0 0      Chaotic 0 0      regime H(1) 1  1  0 1      1    1    Ordered regime 1  1  1  0      0 0      H(0) 1  1  1  1      1    1    Derrida plot is a discrete analogue of the Lyapunov coefficient for continuous t=0 t=1 systems. H(0) = 2 H(1) = 1June 15, 2009 Areejit Samal
  43. 43. Dynamics of the E. coli TRN controlling metabolism is highly ordered in contrast to that of Random Boolean Networks 500 K can be as large as 8 400 H(1) 300 200 100 0 0 100 200 300 400 500 H(0) Reference: S.A. Kauffman (1993) Reference: A. Samal and S. Jain (2008)Kauffman found that Random Boolean Networks (RBN)  The E. coli TRN controlling metabolism has with K=2 are at the edge of chaos using Derrida Plot.  input functions with K=8 also. However, Derrida plot is the discrete analog of the Lyapunov the dynamics of the E. coli TRN is highly coefficient. Derrida plot for RBNs with K>2 are found to  ordered .be above the diagonal and their dynamics is quite chaotic.  June 15, 2009 Areejit Samal
  44. 44. System is far from edge of chaos• The simple architecture of the genetic network controlling E. coli metabolism endows the system with the property of – Homeostasis – Flexibility of response• Note that the dynamics is highly ordered and the system is far from the edge of chaos. It has been argued that the advantage of a system staying close to the edge of chaos lies in its ability to evolvable and be flexible.• We have shown that the real system has an architecture with root control by environmental variables which is highly flexible, evolvable and far from the edge of chaos.• Such an architecture of the regulatory network can also be useful for organisms with different cell types. June 15, 2009 Areejit Samal
  45. 45. Acknowledgement Collaboration Sanjay Jain University of Delhi, India ReferenceJune 15, 2009 Areejit Samal

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