Eyes everywhere…: Eyes everywhere…


Modeling fly phototransduction:how quantitative can one get?: Modeling fly phototransduction: how quantitative can one get?


Limits of modeling?: Limits of modeling? vertebrate phototransduction (rods, cones) insect phototransduction olfaction, taste, etc… Comparative systems biology?


Fly photo-transduction: Fly photo-transduction About the phenomenon Molecular mechanism Phenomenological Model Predictions and comparisons with experiment. Outline:


Compound eyeof the fly: Compound eye of the fly


Fly photoreceptor cell : Fly photoreceptor cell hv 50nm 1.5 mm Na+, Ca2+ Microvillus Rhodopsin


Single photon response in Drosophila: a Quantum Bump: Single photon response in Drosophila: a Quantum Bump Low light Dim flash “All-or-none” response Henderson and Hardie, J.Physiol. (2000) 524, 179


Comparison of a fly with a toad.: Comparison of a fly with a toad. From Hardie and Raghu, Nature 413, (2001) Single photon response: Note different scales directions of current!!


Linearity of macroscopic response: Linearity of macroscopic response hv Linear summation over microvillae


Average QB wave-form: Average QB wave-form A miracle fit: Henderson and Hardie, J.Physiol. (2000) 524, 179 QB aligned at tmax


QB variability: QB variability Peak current Jmax (pA) # events


Multi-photon response: Multi-photon response QB waveform Convolution with latency distribution Macroscopic response = average QB Latency distribution determines the average macroscopic response !!! Fluctuations control the mean !!!


Advantages of Drosophila photo-transduction as a model signaling system:: Advantages of Drosophila photo-transduction as a model signaling system: Input: Photons Output: Changes in membrane potential Single receptor cell preps Drosophila genetics


Molecular mechanism offly phototransduction : Molecular mechanism of fly phototransduction


Response initiation: Response initiation PIP2 High [Na+], [Ca++] Low [Na+], [Ca++] Cast: Rh = Rhodopsin; Gabg = G-protein PIP2 = phosphatidyl inositol-bi-phosphate DAG = diacyl glycerol PLCb = Phospholipase C -beta ; TRP = Transient Receptor Potential Channel DAG Kinase


Positive Feedback: Positive Feedback Intermediate [Ca] facilitates opening of Trp channels and accelerates Ca influx. Ca pump


Negative feedback and inactivation: Negative feedback and inactivation PKC Na+, Ca++ Rh* High [Ca++] Cast: Ca++ acting directly and indirectly e.g. via PKC = Protein Kinase C and Cam = Calmodulin Arr = Arrestin (inactivates Rh* ) High [Ca++] Ca pump


Comparison of early steps: Comparison of early steps From Hardie and Raghu, Nature 413, (2001) Vertebrate Drosophila 2nd messengers cGMP DAG


…and another cartoon: …and another cartoon c. d a. b. From Hardie and Raghu, Nature 413, (2001)


InaD signaling complex: InaD signaling complex InaD PDZ domain scaffold From Hardie and Raghu, Nature 413, (2001)


Speed and space: the issue of localization and confinement.: Speed and space: the issue of localization and confinement. Order of magnitude estimate of activation rates: G* ~ PLC* ~ k [G] ~ 10mm2/s 100 / .3 mm2 > 1 ms-1 Diffusion limit on reaction rate Protein (areal) density ! Fast Enough ! Possible role for InaD scaffold ! However if: ~ 1mm2/s 10 / .3 mm2 =.03ms-1 << 1 ms-1 ! Too Slow !


How “complex” should the modelof a complex network be?: How “complex” should the model of a complex network be?


A naïve model: A naïve model “Input” TRP channel Ca2+ ( G*, PLCb*, DAG) low high


Kinetic equations:: Kinetic equations: Activation stage ( G-protein; PLCb; DAG ): QB “generator” stage ( Trp, Ca++ ): # open channels Positive and negative feedback Ca++ influx via Trp* Ca++ outflow/pump Input (Rh activity)


Feedback Parameterization: Feedback Parameterization Parameterized by the “strength” ga (~ ratio at high/low [Ca]) Characteristic concentration KDa and Hill constant ma Note: this has assumed that feedback in instantaneous…


Null-clines and fixed points: Null-clines and fixed points [Ca]=0 [TRP*]=0 null-cline


Problems with the simple model: Problems with the simple model Model Experiment In response to a step of Rh* activity (e.g. in Arr mutant ) QB current relaxes to zero Ca dynamics is fast rather than slow no “overshoot” Long latency is observed “High” fixed point


Order of magnitude estimate of Ca fluxes: Order of magnitude estimate of Ca fluxes [Ca]dark ~ .2mM [Ca]peak ~ 200mM 1 Ca ion / microvillus 1000 Ca ion / microvillus 30% of 10pA Influx 104 Ca2+ / ms Hence, Ca is being pumped out very fast ~ 10 ms-1 [Ca] is in a quasi-equilibrium Note: m-villus volume ~ 5*10-12 ml


Microvillus as a Ca compartment: Microvillus as a Ca compartment Compare 10 ms-1 with diffusion rate across the microvillus: t -1 ~ Dca / d2 ~ 1 mm2/ms / .0025 mm2 = 400 ms -1 But diffusion along the microvillus: t -1 ~ 1 mm2/ms / 1 mm2 = 1 ms -1 is too slow compared to 10ms-1 50nm 1-2 um Hence it is decoupled from the cell. Note: microvillae could not be > .3mm in diameter, i.e it is possible the diameter is set by diffusion limit Ca++


Slow negative feedback: Slow negative feedback Assume negative feedback is mediated by a Ca-binding protein (e.g. Calmodulin??) Slow relaxation


A more ‘biochemically correct’ model:: A more ‘biochemically correct’ model: Delayed Ca negative feedback F+ F- Feedback Cascade


Stochastic effects: Stochastic effects Gillespie, 1976, J. Comp. Phys. 22, 403-434 see also Bort,Kalos and Lebowitz, 1975, J. Comp. Phys. 17, 10-18 Numbers of active molecules are small ! e.g. 1 Rh*, 1-10 G* & PLC*, 10-20 Trp* Chemical kinetics Master equation Numerical simulation Reaction “shot” noise.


Stochastic simulation: Stochastic simulation Event driven Monte-Carlo simulation a.k.a. Gillespie algorithm Gillespie, 1976, J. Comp. Phys. 22, 403-434 see also Bort,Kalos and Lebowitz, 1975, J. Comp. Phys. 17, 10-18 Numbers of molecules (of each flavor) #Xa(t) are updated #Xa(t) #Xa(t) +/- 1 at times ta,i distributed according to independent Poisson processes with transition rates Ga,+/- . Simulation picks the next “event” among all possible reactions. Note: simulation becomes very slow if some of the Reactions are much faster then others. Use a “hybrid” method.


The model is phenomenological…: The model is phenomenological… Many (most?) details are unknown: e.g. Trp activation may not be directly by DAG, but via its breakdown products; Molecular details of Ca-dependent feedback(s) are not known; etc, etc there’s much to be explained on a qualitative and quantitative level… BUT


Identifying “submodules”: Identifying “submodules” Delayed Ca negative feedback F+ F- Feedback Cascade Key dynamical variables define “Submodules”


Rephrased in a “Modular” form: the “ABC model”: Rephrased in a “Modular” form: the “ABC model” “Input” Channel B ( Rh*, G*) Activator (PLCb*, Dag) (TRP) (Ca-dependent inhibition) Ca++ Ca++ Activator – Buffer – Ca-channel


Quantum Bump generation: Quantum Bump generation 20 60 100 140 180 200 400 0 600 TRP* B*/10 PLC* A Threshold for QB generation A B* (A,B) - “phase” plane High probability of TRP channel opening “INTEGRATE & FIRE” process


What about null-cline analysis?: What about null-cline analysis? Problems: 3 variables A,B,C Stochasticity Discreteness “Ghost” fixed point Generalized “Stochastic Null-cline”


Can one calculate anything?: Can one calculate anything? E.g. estimate the threshold for QB generation: A-1 A A+1 C = 0 1 2 PLC* PLC* Am Am f([Ca]) Positive feedback kicks in once channels open Threshold A = AT such that Prob (AT -> AT +1) = Prob (C=0 -> C=1) NOTE: Better still to formulate as a “first passage” problem


Condition for QB generation: Condition for QB generation PLC* [Ca] 1 2 3 4 0 [Ca] A Prob (C=1 C=2) > Prob (C=1 C=0) Amax~ PLC* * AT AT > AQB ([Ca]) AQB ([Ca]) Bistable region/ Bimodal response Reliable QB generation


Quantum Bump theory versus reality: Quantum Bump theory versus reality Model Experiment Latency histogram Average QB profile


Fitting the data: QB wave-form: Fitting the data: QB wave-form Trp*/Trptot Time (arbs) There is a manifold of parameter values providing good fit for shape !!


So what ???: So what ??? “With 4 parameters I can fit an elephant and with 5 it will wiggle its trunk.” E. Wigner


Non-trivial “architectural” constraints: Non-trivial “architectural” constraints Despite multiplicity of fits, certain constraints emerge: Trp activation must be cooperative Activator intermediate must be relatively stable: “integrate and fire” regime. Negative feedback must be delayed Multiple feedback loops are needed Etc, … Furthermore: Fitting certain relation between parameters: “phenotypic manifold” - the manifold in parameter space corresponding to the same quantitative phenotype.


Many more features to explain quantitatively!: Many more features to explain quantitatively!


Constraining the parameter regime…: Constraining the parameter regime… Help from the data on G-protein hypomorph flies: # of G-proteins reduced by ~100 QB “yield” down by factor of 103 Increased latency (5-fold) Fully non-linear QB with amplitude reduced about two-fold


G-protein hypomorph: G-protein hypomorph Model: Experiment: Single G* and PLC* can evoke a QB !! Reduced yield explained by PLC* deactivating before A reaches the QB threshold Relation between yield reduction and increased latency. # PLC* ~ 5 for WT


What happens in response to continuous activation ?: What happens in response to continuous activation ? e.g. if Rh* fails to deactivate


Persistent Rh* activity Relaxation Oscillator : Persistent Rh* activity Relaxation Oscillator 20 60 100 140 180 100 200 300 400 500 Trp* B* A Unstable Fixed Point (A,B*) phase plane A B*


QB trains: theory versus experiment: QB trains: theory versus experiment Qualitative but not quantitative agreement so far… Model: Arrestin mutant (deficient in Rh* inactivation):


Predicted [Caex] dependence : Predicted [Caex] dependence


Observed external [Ca2+] dependence: Observed external [Ca2+] dependence [Caext] mM Coeff of variation Peak current Henderson and Hardie, J.Physiol. (2000) 524.1, 179


What does one learn from the model?: What does one learn from the model? e.g. Mechanisms/parameters controlling: Threshold for QB generation. QB amplitude fluctuations. Latency. Yield (or response failure rate) Latency distribution. Functional dependences: e.g. dependence of everything on [Ca++ ]ext


Modeling methodology questions: Modeling methodology questions Need an intelligent method of searching the parameter space and of characterizing the parameter manifold ??? How does Evolution search the parameter space? Characterizing the “space of models”?? “Convergence proof”?? Given a model that fits N measurements can we expect that it will fit N+1 (even with additional parameters)? How accurate should a prediction be for us to believe that the model is correct ?? Unique??


Summary and Conclusion: Summary and Conclusion A phenomenological model can explain observations and make numerous falsifiable predictions (especially for the functional dependence on parameters). Insight into HOW the system works from understanding the most relevant parameters and processes. ????? Can one get any insight into WHY the system is constructed the way it is (e.g. vertebrate versus insect) ?????


Acknowledgements: Acknowledgements Alain Pumir, (Inst Non-Lineare Nice, France) Rama Ranganathan (U. Texas,SW Medical School) Anirvan Sengupta, (Rutgers) Peter Detwiler (U. Washington) Sharad Ramanathan (Harvard)