BioControl - Week 6, Lecture 1 Goals of this lecture Large metabolic networks organization Design principles for small genetic modules - Rules based on gene demand - Rules based on error minimization Suggested readings The large-scale organization of metabolic networks H Jeong, B Tombor, R Albert, ZN Oltvai, AL Barabási - Nature, 2000 Some protein interaction networks do not exhibit power law statistics Tanaka, R, Yi, TM and Doyle J - 2005 Design principles for elementary gene circuits, Savageau MA, Chaos 2001 Rules for biological regulation based on error minimization Shinar G, Dekel E, Tlusty T and Alon U, PNAS 2006 Elisa Franco, Caltech 1
Looking for design principles LARGE NETWORK ORGANIZATION MODULES complexity Are there design principles in nature? If so, are they compatible with general engineering principles? If we were to design new networks from scratch, what design principles should we use? Elisa Franco, Caltech 2
Biological networks Metabolic (protein-protein) Genetic Interactions between enzymes and metabolites in the A. thaliana citric acid cycle. Enzymes and metabolites are the red dots and interactions between them are the lines. Systematic mapping of genetic interactions in C. elegans identifies common modifiers of diverse signaling pathways Lehner B, Crombie C, Tischler J, Fortunato A, Fraser AG Nat Genet. 2006 Elisa Franco, Caltech 3
Metabolic networks organization The large-scale organization of metabolic networks H Jeong, B Tombor, R Albert, ZN Oltvai, AL Barabási - Nature, 2000 Random network theory (Erdös-Rény) Scale-free network (power-law) Connectivity (empirically measured) P (k) =k γ Connectivity: Poisson RV <k> <k>k P (k) =e k! Internet, social networks... and metabolic networks? Elisa Franco, Caltech 4
Metabolic networks are scale free The large-scale organization of metabolic networks H Jeong, B Tombor, R Albert, ZN Oltvai, AL Barabási - Nature, 2000 Numerical analysis of IWT database, across 43 organisms Metabolic pathway present if enzymes annotated in genome P(k) derived by histograms S1 S2 E1 = P1 P2 substrates enz-sub complex products E. coli A. fulgidus Nodes: substrates/enz/complexes/ products Links: metabolic reactions in-coming links (substrate=educt) vs out-going links (substrate=product) C. elegans Average over 43 organisms Elisa Franco, Caltech 5
Metabolic networks are scale free The large-scale organization of metabolic networks H Jeong, B Tombor, R Albert, ZN Oltvai, AL Barabási - Nature, 2000 Small-world networks Hubs, conserved across species (4% of all substrates considered) Distance between substrates is bounded Distance is robust wrt random node elimination Archaea Bacteria Eukaryotes Elimination of: M=60 => elimination of 8% of the total substrates in E. coli Elisa Franco, Caltech 6
... aren t they? Some protein interaction data do not exhibit power law statistics Tanaka, R, Yi, TM and Doyle J - 2005 Errors arise in frequency plots (differentiation), that do not arise in cumulative plots. P (X >x)=ce αx p(x) =c(e αx e α(x+1) )=c α e αx Discrete CDF and PDF have analytically compatible formulas, but binning and discrete frequency counting can alter the outcome. Cumulative ranking: exponential Frequency plot: incorrectly appears as a power-law α =0.1, x>10 y=-sort(floor(10*(-1+log(rand(1,n))))) loglog(y,1:n) u=unique(y); L=length(u); f=0*u; for k=1:l, f(k)=sum(y==u(k)); end; loglog(u,f) Elisa Franco, Caltech 7
Hub genes experimentally found in eukaryotes Systematic mapping of genetic interactions in C. elegans identifies common modifiers of diverse signaling pathways Lehner B et al Nat Genet. 2006 Genetic networks seem to follow a scale-free node distribution in C. elegans Method: inhibit genes with RNAi target genes: transcription/signaling in development e.g. EGF/MAPK pathway Results: The study revealed new interactions of the EGF pathway Few hub genes, related to chromatin modification/ organization Hubs may be important to understand genetic diseases Elisa Franco, Caltech 8
Genetic networks: levels of design Design principles for elementary gene circuits, Savageau MA, Chaos 2001 Transcription unit: fan in/fan out of signals Modulators + Promoter + Genes + Terminator M1 M2 P G1 G2 T Mode of control: Positive - genes in high demand Negative - genes in low demand Verified in E. coli catabolic systems: - positive control if nutrient is common - negative control if nutrient is rare - antagonistic functions (e.g. biosynthesis and degradation) : opposite regulations - aligned functions (e.g. transport layers): same regulatory mode Input signaling: extra-cellular signals or network signals transmitted to the transcription unit Elisa Franco, Caltech 9
Genetic networks: levels of design Design principles for elementary gene circuits, Savageau MA, Chaos 2001 Logic unit: number of modulator sites and their logical combination M1 M2 P G Expression cascade: Gene->mRNA->Protein->Metabolite R1 R2 AND yes no OFF yes no ON no yes OFF no no OFF Connectivity: inputs and outputs of transcription units -> for coordinated gene expression - Autogenous regulation - Elementary gene circuits - motifs - E. coli: 2-3 inputs per transcription unit, some t.u. have up to 50 outputs - Operons (polycistronic mrna) - Regulons (separate txn units with same regulatory domain) Elisa Franco, Caltech 10
Comparing different designs Design principles for elementary gene circuits, Savageau MA, Chaos 2001 Deterministic continuous models are (usually) a good choice to gain intuition on design principles. Suggested method: Restrict differences to a single specific process Models are internally equivalent Parameters associated with the different process (to be compared) are free Reduce degrees of freedom by external equivalence Eliminate arbitrary model differences If symbolic solutions to the differential equations are available, no numerical analysis is required. Elisa Franco, Caltech 11
Comparing different designs Design principles for elementary gene circuits, Savageau MA, Chaos 2001 Generalized mass action models (GMA) allow in some cases explicit solutions dx i dt = k j complexes producing Xi α ik X g ijk j complexes in which Xi is a reactant β ik X h ijk j k j Synergistic (S-)systems dx i = α i X g ij j β i dt j Explicit solution for equilibria Explicit conditions for local stability j X h ij j Logarithmically differentiable functions IDEA for local linear analysis: Taylor series expansion: log Z = f(log X) log Z = log Z 0 + j log Z log X j Z 0 log X j X j0 In cartesian coordinates: (exponentiating) Z = a j X α j j Elisa Franco, Caltech 12
Example Design principles for elementary gene circuits, Savageau MA, Chaos 2001 Study g13, g43 (inducer) and g15, g45 (regulator). Other parameters held equal (internal equiv.) Steady states compatible for all possible models (external equiv.) mrna Enzyme Inducer mrna Regulator Elisa Franco, Caltech 13
Rules for biological regulation based on error minimization Shinar G, Dekel E, Tlusty T and Alon U, PNAS 2006... not just demand of genes (Savageau) TF Transcriptional control Activator or inhibitor M P G If transcription factor not bound to its site => Site exposed to nonspecific binding Binding of inactive TF Cross-talks from other TFs Lateral gene transfer... => ERRORS: reduction of fitness: error-load p - fraction of time a gene is expressed / in demand Fitness reduction Repressor E R = p f 1 1- high expression state Activator E A =(1 p) f 0 0- low expression state Elisa Franco, Caltech 14
Rules for biological regulation based on error minimization Repressors are advantageous when the demand is lower than a threshold: p< 1 1+ f 1 f 0 Replacement of mode of control due to selection pressure after random mutation E coli selection threshold s min 10 8 10 7 minimum fitness advantage for a mutant to take over repressor fixation activator fixation E A E R >s min E R E A >s min Elisa Franco, Caltech 15
Summary Networks - design principles - Metabolic and genetic networks seem to be scale-free Genetic modules and design of regulatory interaction - Rules for demand of gene expression - Rules for error minimization NEXT LECTURE - Network motifs - Dynamic role of motifs Elisa Franco, Caltech 16