Lecture 2: Analysis of Biomolecular Circuits Richard M. Murray Caltech CDS/BE Goals: Give a short overview of the control techniques applied to biology - uncertainty management - system identification - design of dynamics - (fundamental limits of performance) References D. Del Vecchio and R. M. Murray, Biomolecular eedback Systems, 29. http://www.cds.caltech.edu/~murray/amwiki/bfs 29 Asian Control Conference (ASCC), 26 August 29
Control Theory for Biological Systems What s different about biological systems Complexity - biological systems are much more complicated than engineered systems Communications - signal representations are very different (spikes, proteins, etc) Uncertainty - very large uncertainty in components; don t match current tools Evolvability - mutation, selection, etc Rao, Kirby and Arkin PLoS Biology, 24 Potential application areas for control tools System ID - what are the appropriate component abstractions and models? Analysis - what are key biological feedback mechanisms that lead to robust behavior? Design - how to we (re-)design biological systems to provided desired function? undamental limits - what are the limits of performance and robustness for a given biological network topology? ASCC, Aug 9 2
Example: Self-Repression (from Astrom & Murray 8) Common motif in transcriptional regulation A binds to its own promotor region If A is bound, represses binding of RNA polymerase Use #: decrease sensitivity to input disturbances requency responses demonstrates effect dm dt G ol pv(s) = G cl pv(s) = = α(p) γm v, dp β (s + γ)(s + δ) dt = βm δp β (s + γ)(s + δ)+βσ, σ = nα kp n e ( + kp n e ) 2 Use #2: use to decrease response time Use a stronger promotor to increase low freq gain Increases bandwidth faster response ASCC, Aug 9 3
System Identification Using Cell Noise Traditional systems identification Engineering: forced response. Difficult to do in in vivo (eg, sinusoids are tricky) Biology: gene knockouts; steady state measurements using gene arrays Idea: use noise as a forcing function Steady state distributions are not enough if extrinsic noise is present Need to use correlation data instead ASCC, Aug 9 4
System ID of a Synthetic Circuit (Dunlop, Elowitz & M) E. coli Plasmid (7,5 base pairs) ~ copies per cell Chromosome (4,6, base pairs) copy per cell Results to date Synthetic circuit demonstrates viability of approach Implemented on natural circuit (using promoter fusion) ASCC, Aug 9
E igure 5 CA igure 5 C E A camp-crp camp-crp C E X Y Y Z Xinducer inducer Y camp-crp camp-crp ASCC, Aug 9 camp-crp Z Z camp-crp camp-crp camp-crp camp-crp camp-crp B B System ID of an in vivo circuit R Y,Z (!) R R Y,Z (!) Y,Z (!) P P gals - YP gals - YP P P gale - CP gale - CP P P gals - YP gals - YP P P gale - CP gale - CP P gals - YP P P gale - CP gals - YP P gale - CP DD!6!4!2 2 4 6!6!4!2 2 4 6!.5 without inducer with inducer D.5.5.5.5.5 D!.5 without inducer!.5 without inducer inducer!.5 without inducer inducer!.5 with inducer!6!4!2 2!6!4!.5!2 2.5 4 6 4 6!!6!4!2 (mins) (mins) 2 4 6! (mins) P gals.5.5 - YP P gale - CP R YP,CP (!) P gals - YP P gale - CP!.5!.5.5.5!.5!.5 P gals - YP P gale - CP R Y,Z!.5.5 Galactose regulation in E. coli GalE regulated by CRP via a feedforward loop represses mm feedforward loop when mm mm is present Promoter mm fusions measure and GalE concentrations!6!4!2 2 4 6!6!4!2 2 4 6!.5!6!4!2 2 4 6!6!4!2 2 4 6 (mins)!6!4!2 2 4 6!6!4!2 2 4 6!6!4!2 2 4 6 System ID shows mm L is not active mm Addition of shows no change in correlations => is not actively regulating GalE!6 mm!4!2 2 4 6 mm mm.5 Hypothesis: repression dominant If repression by is large, mm is always off => no connection mm Removal of recovers expected correlations!6!4!2 2 4 6 mm mm mm!.5 mm mm!6!4!2 2 4 6 6
Improving the Performance of Oscillators Toggelator Coupled oscillators Add additional delay (ACi) Ugander, Dunlop & M, ACC 7 ASCC, Aug 9 7
Possible ramework for Analysis and Design external disturbances Σ N ( ) N 2 ( ).. N n ( ) N n2 ( ) Nonlinear Coupling e τ s... P (s) Delay Matrix Unmodeled Dynamics... P n (s) System Dynamics e τ ms L Interconnection Matrix external disturbances Σ Analysis via loop gain Design the easy parts Interconnection matrix Time delay matrix Design tools exist for pairwise combinations Linear + uncertain = robust control theory Linear + nonlinear = describing functions Linear + network = formation stabilization Linear + delay = loquet analysis Open questions What is the class of feedback compensators we can obtain using L and τ? How do we specify robustness and performance in highly stochastic settings? Can feedback be used to design robust dynamics that implements useful functionality? ASCC, Aug 9 8