Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto...

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Visual Transduction: A Paradigm for Signaling E. DiBenedetto [email protected] Department of Mathematics, Vanderbilt University Nashville, TN 37240, USA UAB April 8 2014 – p.1/37

Transcript of Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto...

Page 1: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Visual Transduction: A Paradigmfor Signaling

E. [email protected]

Department of Mathematics, Vanderbilt University

Nashville, TN 37240, USA

UAB April 8 2014 – p.1/37

Page 2: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Micrograph of Salamander and Frog Retina

UAB April 8 2014 – p.2/37

Page 3: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Micrograph of the Salamander Rod Outer Segment

Thicknessσεo of outer shell comparable to distanceνεo between any two

discs. For the Salamanderεo ≈ 14 nm and ν ≈ σ ≈ 1. Here εo is the

thickness of the “layers/discs”. UAB April 8 2014 – p.3/37

Page 4: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Geometry of the Salamander ROS

H ≈ 22µm R ≈ 5.5µm εo ≈ 14 nm

ν ≈ 1 σ ≈ 1 n ≈ 800

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DR

H

Cjo

CjF+

j

Lj

2(R + σε)

photons

UAB April 8 2014 – p.4/37

Page 5: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Geometry of the ROS with Incisures

Sε =DR+σε − DR

× (0, H) Outer Shell ε ≤ εo

Ij = interdiscal spaces thickness ≈ νε ν ≈ 1

Bε = Vε × (0, H) blade-like incisures n ≈ 800

.......................................................................................................................................................................................................................................................................................................................... ................................................................ ........................................................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .............................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .............. ...................................................................................................

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............................................................................................................... ......................................................................... ......................................................................... .........................................................................DRH Cj

Cj F+jLj

2(R+ ")photons

generi dis Cj of upper andlower fa es Fj and lateralboundary Lj

UAB April 8 2014 – p.5/37

Page 6: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Incisures in Frog ROS

Vision Res. Vol. 22, 1982, pp. 1417–1428

UAB April 8 2014 – p.6/37

Page 7: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Incisures in Rat ROS

Vision Res. Vol. 7, 1967, pp. 829–836UAB April 8 2014 – p.7/37

Page 8: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Vision-I

• Geometry/Anatomy is part of the System

Behavior of a Biological Process

• Intricate Geometry Fosters Novel

Mathematics

UAB April 8 2014 – p.8/37

Page 9: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Light Activation

UAB April 8 2014 – p.9/37

Page 10: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Signaling Cascade on the Activated Disc

UAB April 8 2014 – p.10/37

Page 11: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Cascade and Resulting Photocurrent

UAB April 8 2014 – p.11/37

Page 12: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Cascade and Resulting Photocurrent

Photon =⇒ R =⇒ R∗ =⇒ G =⇒ G∗ =⇒ PDE=⇒ PDE∗

depletion of cGMP=⇒ closing of the channels.

UAB April 8 2014 – p.11/37

Page 13: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Cascade and Resulting Photocurrent

Photon =⇒ R =⇒ R∗ =⇒ G =⇒ G∗ =⇒ PDE=⇒ PDE∗

depletion of cGMP=⇒ closing of the channels.

The resultingphotocurrentgenerated by the cGMP–gatedchannels is

JcG = jmaxcG

[cGMP]mcG

KmcGcG + [cGMP]mcG

([cGMP] = u(x, t); mcG = 2

).

UAB April 8 2014 – p.11/37

Page 14: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Cascade and Resulting Photocurrent

Photon =⇒ R =⇒ R∗ =⇒ G =⇒ G∗ =⇒ PDE=⇒ PDE∗

depletion of cGMP=⇒ closing of the channels.

The resultingphotocurrentgenerated by the cGMP–gatedchannels is

JcG = jmaxcG

[cGMP]mcG

KmcGcG + [cGMP]mcG

([cGMP] = u(x, t); mcG = 2

).

NOTE: R, R∗, G, G∗, PDE, PDE∗, GC live on a "surface",

whereascGMP and Ca2+ live in a "volume".

UAB April 8 2014 – p.11/37

Page 15: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Vision-II

• Combining Biophysics, Biochemistry and

Geometry/Anatomy

• (subtitle) System Behavior

UAB April 8 2014 – p.12/37

Page 16: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Revisiting the Process by Injecting 3 New Ideas

UAB April 8 2014 – p.13/37

Page 17: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Revisiting the Process by Injecting 3 New Ideas

• Modeling on Site (Surface to Volume)

PDE∗–cGMP interactions should be modeled as fluxes

sources on the disc surfaces. Similarly for Ca2+ fluxes on

the outer shell.

UAB April 8 2014 – p.13/37

Page 18: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Revisiting the Process by Injecting 3 New Ideas

• Modeling on Site (Surface to Volume)

PDE∗–cGMP interactions should be modeled as fluxes

sources on the disc surfaces. Similarly for Ca2+ fluxes on

the outer shell.

• Modeling by the Geometry

Domain of diffusion ishorizontally thin(interdiscal space)

andvertically thin(outer shell). Account for both and their

interaction.

UAB April 8 2014 – p.13/37

Page 19: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Revisiting the Process by Injecting 3 New Ideas

• Modeling on Site (Surface to Volume)

PDE∗–cGMP interactions should be modeled as fluxes

sources on the disc surfaces. Similarly for Ca2+ fluxes on

the outer shell.

• Modeling by the Geometry

Domain of diffusion ishorizontally thin(interdiscal space)

andvertically thin(outer shell). Account for both and their

interaction.

• Homogenizing and Concentrating

Reduce the problem to one with simpler geometry, still

preserving thesamefeatures of the original one.

UAB April 8 2014 – p.13/37

Page 20: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Limit with Constant Volume Fraction

The ratio of the volume occupied by the discsCε,j to the volumeof Ω − Bε is kept constant asε → 0.

vol(⋃n

j = 1 Cε,j

)

vol(Ω − Bε)= µo

.......................................................................................................................................................................................................................................................................................................................... ................................................................ ........................................................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .............................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .............. ...................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ...............

.....................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................ .............. ............................................................................................

......................................................................................................................................................................................................................................................................................................................................................

............................................................................................................... ......................................................................... ......................................................................... .........................................................................DRH Cj

Cj F+jLj

2(R+ ")photons

generi dis Cj of upper andlower fa es Fj and lateralboundary Lj

UAB April 8 2014 – p.14/37

Page 21: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Formal Limiting Domains-I

Vε −→ V = ro < x1 < R × x2 = 0DR+σε − Vε −→ DR − VDR+σε − DR −→ ∂DR

...................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................................................................................... .........................................................................................................

.........................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................... ................................................................................................................................................................................................................. ................................................................................................................................................................................................................. roro O O

x1 x1x2x2

UAB April 8 2014 – p.15/37

Page 22: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Formal Limiting Domains-II

Sε = DR+σε − DR × (0, H) −→ S = ∂DR × (0, H)

Bε = Vε × (0, H) −→ V × (0, H)

Ωε = DR+σε − Vε × (0, H) −→ Ω = DR − V × (0, H)

....................................................................................................................................................................

..................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................................................................................................................

..........................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................. ...................................................................................................... ............................................................................................................................................................................................................................................................................................................................... ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. .............................................................................................................................................................. ..............

....................................................................................................................... .................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................

....................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .................................................................................................................................... .............................................................................................................................................................. ..............

.......................................................................................................................zz

x1x1 x2x2

UAB April 8 2014 – p.16/37

Page 23: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Limiting Domains and Functions asε → 0

The approximating 2nd MessengerscGMPandCa2+ (sayuε for

cGMP)tend to 3 functions each satisfying a diffusion process in

its domain of definition.

UAB April 8 2014 – p.17/37

Page 24: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Limiting Domains and Functions asε → 0

The approximating 2nd MessengerscGMPandCa2+ (sayuε for

cGMP)tend to 3 functions each satisfying a diffusion process in

its domain of definition.

• u(x, y, z, t) defined in the limiting cytoplasm and called the

interior limit

UAB April 8 2014 – p.17/37

Page 25: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Limiting Domains and Functions asε → 0

The approximating 2nd MessengerscGMPandCa2+ (sayuε for

cGMP)tend to 3 functions each satisfying a diffusion process in

its domain of definition.

• u(x, y, z, t) defined in the limiting cytoplasm and called the

interior limit

• uS defined inS and called thelimit at the outer shell

UAB April 8 2014 – p.17/37

Page 26: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Limiting Domains and Functions asε → 0

The approximating 2nd MessengerscGMPandCa2+ (sayuε for

cGMP)tend to 3 functions each satisfying a diffusion process in

its domain of definition.

• u(x, y, z, t) defined in the limiting cytoplasm and called the

interior limit

• uS defined inS and called thelimit at the outer shell

• uB defined in the limiting bladeB, and called thelimit inthe incisure

UAB April 8 2014 – p.17/37

Page 27: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Global Weak Form

(1 − µo)

∫∫

ΩT−BT

utϕ + ∇xu · ∇xϕ + (u − f)ϕ

dxdt

interior

+σεo

∫∫

ST

uS,tϕ + ∇SuS · ∇Sϕ

dηdt

outer shell

+2

∫∫

BT

rθεo(r)

uB,tϕ + ∇BuB · ∇Bϕ

drdzdt

incisure

= 0

valid for all testing functionsϕ ∈ W 1,2(ΩT ) roughly speaking in

the dual classes ofu, uB anduS.

UAB April 8 2014 – p.18/37

Page 28: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Vision-III

• Novel Mathematics

• (subtitle) Partial Differential Equations in

Fractured Domains or Domains with Spikes

UAB April 8 2014 – p.19/37

Page 29: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The form of f(u) Contains the Biochemistry

UAB April 8 2014 – p.20/37

Page 30: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The form of f(u) Contains the Biochemistry

f([cGMP]) = 12

βdark[cGMP]− α

1 + ([Ca2+]/K cyc)mcyc

− δz∗k∗σ;hyd[PDE∗]σ[cGMP]

Here[cGMP] and[Ca2+] are unknowns to be determined by the

system of equations.[PDE∗] is adatum generated by the

activation cascade. More on this later.

UAB April 8 2014 – p.20/37

Page 31: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The form of f(u) Contains the Biochemistry

f([cGMP]) = 12

βdark[cGMP]− α

1 + ([Ca2+]/K cyc)mcyc

− δz∗k∗σ;hyd[PDE∗]σ[cGMP]

Here[cGMP] and[Ca2+] are unknowns to be determined by the

system of equations.[PDE∗] is adatum generated by the

activation cascade. More on this later.

VISION: Inform Mathematics with Biochemistry

UAB April 8 2014 – p.20/37

Page 32: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Simulated and Experimental Current Drop 1 − J/jdark

UAB April 8 2014 – p.21/37

Page 33: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Simulated and Experimental Current Drop 1 − J/jdark

• Experimental data for mouse SPR provided by F. Rieke

(Seattle).

UAB April 8 2014 – p.21/37

Page 34: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Simulated and Experimental Current Drop 1 − J/jdark

• Experimental data for mouse SPR provided by F. Rieke

(Seattle).

0 0.5 1 1.50

2

4

6

8

10

t [s]

1−j to

t(t)/j da

rk [%

]

experimentsimulation

UAB April 8 2014 – p.21/37

Page 35: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Simulated and Experimental Current Drop 1 − J/jdark

• Experimental data for mouse SPR provided by F. Rieke

(Seattle).

0 0.5 1 1.50

2

4

6

8

10

t [s]

1−j to

t(t)/j da

rk [%

]

experimentsimulation

• Full Model (non homogenized) 8sec simulation on 20

processors of IBM series cluster (Oak Ridge Labs) took 57

hours. Same 8sec simulation with theHM (homogenized

model) 8sec simulation took 50 seconds of an Intel Xeon

3GHz.

UAB April 8 2014 – p.21/37

Page 36: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Single Photon. Current as a Function of the Incisures

(Salamander)

UAB April 8 2014 – p.22/37

Page 37: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Single Photon. Current as a Function of the Incisures

(Salamander)

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time [s]

1−j(t

)/j da

rk [

%]

A01361423

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time [s]

1−j(t

)/j da

rk [

%]

B01361423

UAB April 8 2014 – p.22/37

Page 38: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Single Photon. Current as a Function of the Incisures

(Salamander)

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time [s]

1−j(t

)/j da

rk [

%]

A01361423

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time [s]

1−j(t

)/j da

rk [

%]

B01361423

• Continuous line:tridimensional homogenized model; Dashed line: longitudinal

well-stirred model; Dotted line: totally well-stirred model.

UAB April 8 2014 – p.22/37

Page 39: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Single Photon. Current as a Function of the Incisures

(Salamander)

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time [s]

1−j(t

)/j da

rk [

%]

A01361423

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time [s]

1−j(t

)/j da

rk [

%]

B01361423

• Continuous line:tridimensional homogenized model; Dashed line: longitudinal

well-stirred model; Dotted line: totally well-stirred model.

• Panel A: Each incisure is an isosceles triangle of base15nm and height4.64µm.

There are 23 of them to reproduce total area of≈ 0.82µm2 estimated by

Olson-Pugh, 1994.

UAB April 8 2014 – p.22/37

Page 40: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Single Photon. Current as a Function of the Incisures

(Salamander)

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time [s]

1−j(t

)/j da

rk [

%]

A01361423

0 0.2 0.4 0.6 0.8 1 1.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time [s]

1−j(t

)/j da

rk [

%]

B01361423

• Continuous line:tridimensional homogenized model; Dashed line: longitudinal

well-stirred model; Dotted line: totally well-stirred model.

• Panel A: Each incisure is an isosceles triangle of base15nm and height4.64µm.

There are 23 of them to reproduce total area of≈ 0.82µm2 estimated by

Olson-Pugh, 1994.

• Panel B: Incisures with equal area but half the length and double the width of

those in the left figure (wide and short, with tip away from activation site).

UAB April 8 2014 – p.22/37

Page 41: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Vision-IV

• Numerical Analysis, Programming,

Simulating

• (subtitle) Capability of Replacing Wet-Bench

Experiments with Virtual Experiments

UAB April 8 2014 – p.23/37

Page 42: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Activation/Deactivation Cascade, Variability and

Fidelity of the SPR

UAB April 8 2014 – p.24/37

Page 43: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Activation/Deactivation Cascade, Variability and

Fidelity of the SPR

0 0.2 0.4 0.6 0.8 1 1.2

0

0.5

1

1.5

t [s]

1−j(t

)/j da

rk [%

]

µµ+σµ−σσ/µ

0 0.2 0.4 0.6 0.8

0

1

2

3

4

5

6

7

8

t [s]

1−j(t

)/j da

rk [%

]

µµ+σµ−σσ/µ

UAB April 8 2014 – p.24/37

Page 44: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Activation/Deactivation Cascade, Variability and

Fidelity of the SPR

0 0.2 0.4 0.6 0.8 1 1.2

0

0.5

1

1.5

t [s]

1−j(t

)/j da

rk [%

]

µµ+σµ−σσ/µ

0 0.2 0.4 0.6 0.8

0

1

2

3

4

5

6

7

8

t [s]

1−j(t

)/j da

rk [%

]

µµ+σµ−σσ/µ

• Left Panel: Experimental SPR responses for Salamander (F. Rieke). Each curve

is the average of a few hundred experiments. Thick black curve is their average.

Red curves are simulations.

UAB April 8 2014 – p.24/37

Page 45: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Activation/Deactivation Cascade, Variability and

Fidelity of the SPR

0 0.2 0.4 0.6 0.8 1 1.2

0

0.5

1

1.5

t [s]

1−j(t

)/j da

rk [%

]

µµ+σµ−σσ/µ

0 0.2 0.4 0.6 0.8

0

1

2

3

4

5

6

7

8

t [s]

1−j(t

)/j da

rk [%

]

µµ+σµ−σσ/µ

• Left Panel: Experimental SPR responses for Salamander (F. Rieke). Each curve

is the average of a few hundred experiments. Thick black curve is their average.

Red curves are simulations.

• Coefficient of VariationCV=σ/µ; standard deviationσ over the meanµ

UAB April 8 2014 – p.24/37

Page 46: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Activation/Deactivation Cascade, Variability and

Fidelity of the SPR

0 0.2 0.4 0.6 0.8 1 1.2

0

0.5

1

1.5

t [s]

1−j(t

)/j da

rk [%

]

µµ+σµ−σσ/µ

0 0.2 0.4 0.6 0.8

0

1

2

3

4

5

6

7

8

t [s]

1−j(t

)/j da

rk [%

]

µµ+σµ−σσ/µ

• Left Panel: Experimental SPR responses for Salamander (F. Rieke). Each curve

is the average of a few hundred experiments. Thick black curve is their average.

Red curves are simulations.

• Coefficient of VariationCV=σ/µ; standard deviationσ over the meanµ

• Center Panel:Experimental CV for the Salamander SPR (Rieke)

UAB April 8 2014 – p.24/37

Page 47: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Activation/Deactivation Cascade, Variability and

Fidelity of the SPR

0 0.2 0.4 0.6 0.8 1 1.2

0

0.5

1

1.5

t [s]

1−j(t

)/j da

rk [%

]

µµ+σµ−σσ/µ

0 0.2 0.4 0.6 0.8

0

1

2

3

4

5

6

7

8

t [s]

1−j(t

)/j da

rk [%

]

µµ+σµ−σσ/µ

• Left Panel: Experimental SPR responses for Salamander (F. Rieke). Each curve

is the average of a few hundred experiments. Thick black curve is their average.

Red curves are simulations.

• Coefficient of VariationCV=σ/µ; standard deviationσ over the meanµ

• Center Panel:Experimental CV for the Salamander SPR (Rieke)

• Right Panel: Experimental CV for the Mouse SPR (Rieke)

UAB April 8 2014 – p.24/37

Page 48: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Activation Cascade and Quenching

UAB April 8 2014 – p.25/37

Page 49: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Activation Cascade and Quenching

• A molecule of rhodopsin, upon activation at some random

locationxo, follows its random patht → x(t) in Deff and

becomes inactivated abruptly after a random timetR, of

averageτR.

UAB April 8 2014 – p.25/37

Page 50: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Activation Cascade and Quenching

• A molecule of rhodopsin, upon activation at some random

locationxo, follows its random patht → x(t) in Deff and

becomes inactivated abruptly after a random timetR, of

averageτR.

• During the random interval[0, tR) R∗ undergoesn

molecular statesR∗

j , j = 1, . . . , n, each with

transducer-activation rateνj.

UAB April 8 2014 – p.25/37

Page 51: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Activation Cascade and Quenching

• A molecule of rhodopsin, upon activation at some random

locationxo, follows its random patht → x(t) in Deff and

becomes inactivated abruptly after a random timetR, of

averageτR.

• During the random interval[0, tR) R∗ undergoesn

molecular statesR∗

j , j = 1, . . . , n, each with

transducer-activation rateνj.

• For exampleR∗

j might be the phosphorylated states ofR∗,

identified by the numberj of phosphates attached toR∗ by

RK. The state(n + 1) is identified withR∗ quenching.

UAB April 8 2014 – p.25/37

Page 52: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

The Activation Cascade and Quenching

• A molecule of rhodopsin, upon activation at some random

locationxo, follows its random patht → x(t) in Deff and

becomes inactivated abruptly after a random timetR, of

averageτR.

• During the random interval[0, tR) R∗ undergoesn

molecular statesR∗

j , j = 1, . . . , n, each with

transducer-activation rateνj.

• For exampleR∗

j might be the phosphorylated states ofR∗,

identified by the numberj of phosphates attached toR∗ by

RK. The state(n + 1) is identified withR∗ quenching.

• The transitions from the statej to (j + 1) occurs at some

random time0 < tj ≤ tn = tR whereto = 0.UAB April 8 2014 – p.25/37

Page 53: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Variability Versus Biochemistry

UAB April 8 2014 – p.26/37

Page 54: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Variability Versus Biochemistry

• If CV ≈ 1/√

n (Poisson statistics), then one needs about20

Biochemical steps to quenching

UAB April 8 2014 – p.26/37

Page 55: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Variability Versus Biochemistry

• If CV ≈ 1/√

n (Poisson statistics), then one needs about20

Biochemical steps to quenching

• The Biochemistry says that there are only 6 phosphorylation

sites on the C-terminus of the rhodopsin

UAB April 8 2014 – p.26/37

Page 56: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Variability Versus Biochemistry

• If CV ≈ 1/√

n (Poisson statistics), then one needs about20

Biochemical steps to quenching

• The Biochemistry says that there are only 6 phosphorylation

sites on the C-terminus of the rhodopsin

• How to solve the puzzle ?

UAB April 8 2014 – p.26/37

Page 57: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Variability Versus Biochemistry

• If CV ≈ 1/√

n (Poisson statistics), then one needs about20

Biochemical steps to quenching

• The Biochemistry says that there are only 6 phosphorylation

sites on the C-terminus of the rhodopsin

• How to solve the puzzle ?

• Pugh, E. N. Jr.,Variability of Single Photon Responses: A Cut in the

Gordian Knot of Rod Phototransduction ?Neuron, 1999, 23:205–208

UAB April 8 2014 – p.26/37

Page 58: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Vision-V

• Interplay Between Stochastic and

Deterministic Processes. Both are Present

Essentially Always

• (subtitle) Analyze Randomness by Knowing

the Biochemistry

• (subtitle) Again System Biology

UAB April 8 2014 – p.27/37

Page 59: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Choosing a Biochemistry

UAB April 8 2014 – p.28/37

Page 60: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Choosing a Biochemistry

• Phosphorylation of rhodopsin by RK is a cooperative

process, i.e., the incorporation of one or two phosphates

increases the probability of further phosphorylation.

• The catalytic activity ofR∗ decreases with increasing levelsof phosphorylation, at a rate of about 12% for eachadditional level of phosphorylation.

• V. Gurevich, The Selectivity of Visual Arrestin for Light-Activated

Rhodopsin is Controlled by Multiple Non-Redundant Mechanisms,J.

Biol. Chem,1998, 15501–15506.

UAB April 8 2014 – p.28/37

Page 61: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Continuous Time Markov Chain (CTMC):

Variability and Biochemistry

UAB April 8 2014 – p.29/37

Page 62: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Continuous Time Markov Chain (CTMC):

Variability and Biochemistry

UAB April 8 2014 – p.29/37

Page 63: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Continuous Time Markov Chain (CTMC):

Variability and Biochemistry

Hereλj are affinities to phosphorylation andµj are affinities to

Arrestin (Arr). Bernoulli process with probabilitiesλj

λj+µjand

µj

λj+µj.

UAB April 8 2014 – p.29/37

Page 64: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Continuous Time Markov Chain (CTMC):

Variability and Biochemistry

Hereλj are affinities to phosphorylation andµj are affinities to

Arrestin (Arr). Bernoulli process with probabilitiesλj

λj+µjand

µj

λj+µj.

The λj and µj represent the Biochemistry of Deactivation.

UAB April 8 2014 – p.29/37

Page 65: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Simulations and Functionals

UAB April 8 2014 – p.30/37

Page 66: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Simulations and Functionals

Random events are(a) Activation site,(b) Brownian path of R∗,

(c) Shutoff ofR∗ in n random steps.

UAB April 8 2014 – p.30/37

Page 67: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Simulations and Functionals

Random events are(a) Activation site,(b) Brownian path of R∗,

(c) Shutoff ofR∗ in n random steps. A sample of the Variability

Functionals:

I∗(t) total current up to timet

E∗∗(t) total activity of [PDE∗] up to timet

UAB April 8 2014 – p.30/37

Page 68: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Simulations and Functionals

Random events are(a) Activation site,(b) Brownian path of R∗,

(c) Shutoff ofR∗ in n random steps. A sample of the Variability

Functionals:

I∗(t) total current up to timet

E∗∗(t) total activity of [PDE∗] up to timet

Main issue is:

• Why is the CV of the photocurrent low ?

UAB April 8 2014 – p.30/37

Page 69: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Simulated and Experimental CV(= σ/µ) of

Current Drop

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

t [s]

Q =

σ/µ

Coefficient of Variability for Salamander

simulationexperiment

Initial instability due to Incisures; up to 32 for the Salamander,

and only 1 for MouseUAB April 8 2014 – p.31/37

Page 70: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Simulated CV(= σ/µ). Mouse

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

CV

of E

**(t

)

A NN

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

CV

of I

*(t)

B NN

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

t [s]

CV

of E

**(t

)

C EE

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

t [s]

CV

of I

*(t)

D EE

n=2n=3n=4n=5

UAB April 8 2014 – p.32/37

Page 71: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Simulated CV(= σ/µ). Salamander

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

CV

of E

**(t

)

A NN

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

CV

of I

*(t)

B NN

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

t [s]

CV

of E

**(t

)

C EE

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

t [s]

CV

of I

*(t)

D EE

n=2n=3n=4n=5

UAB April 8 2014 – p.33/37

Page 72: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Conclusions

• Homogenization and Concentrated Capacity permits one to extract

functional properties of the cell architecture. In particular:

UAB April 8 2014 – p.34/37

Page 73: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Conclusions

• Homogenization and Concentrated Capacity permits one to extract

functional properties of the cell architecture. In particular:

• Permits one to identify the role of the incisures; an open issue in

Biology up to now since they are not accessible to experiments.

UAB April 8 2014 – p.34/37

Page 74: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Conclusions

• Homogenization and Concentrated Capacity permits one to extract

functional properties of the cell architecture. In particular:

• Permits one to identify the role of the incisures; an open issue in

Biology up to now since they are not accessible to experiments.

• Separates the process intoactivation anddiffusion of the secondmessengers cGMP and Ca2+ in the cytoplasm. These are, as single

modules, not accessible to experiments.

UAB April 8 2014 – p.34/37

Page 75: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Conclusions

• Homogenization and Concentrated Capacity permits one to extract

functional properties of the cell architecture. In particular:

• Permits one to identify the role of the incisures; an open issue in

Biology up to now since they are not accessible to experiments.

• Separates the process intoactivation anddiffusion of the secondmessengers cGMP and Ca2+ in the cytoplasm. These are, as single

modules, not accessible to experiments.

• Analysis of the Activation cascade and experimental results, drive the

choice of the Biochemistry of the shutoff mechanism of the receptor.

They also permit to separate the CV of the activation stage and that of the

photocurrent. In particular:

UAB April 8 2014 – p.34/37

Page 76: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Conclusions

• Homogenization and Concentrated Capacity permits one to extract

functional properties of the cell architecture. In particular:

• Permits one to identify the role of the incisures; an open issue in

Biology up to now since they are not accessible to experiments.

• Separates the process intoactivation anddiffusion of the secondmessengers cGMP and Ca2+ in the cytoplasm. These are, as single

modules, not accessible to experiments.

• Analysis of the Activation cascade and experimental results, drive the

choice of the Biochemistry of the shutoff mechanism of the receptor.

They also permit to separate the CV of the activation stage and that of the

photocurrent. In particular:

• CV is high at the activation stage

UAB April 8 2014 – p.34/37

Page 77: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Conclusions

• Homogenization and Concentrated Capacity permits one to extract

functional properties of the cell architecture. In particular:

• Permits one to identify the role of the incisures; an open issue in

Biology up to now since they are not accessible to experiments.

• Separates the process intoactivation anddiffusion of the secondmessengers cGMP and Ca2+ in the cytoplasm. These are, as single

modules, not accessible to experiments.

• Analysis of the Activation cascade and experimental results, drive the

choice of the Biochemistry of the shutoff mechanism of the receptor.

They also permit to separate the CV of the activation stage and that of the

photocurrent. In particular:

• CV is high at the activation stage

• CV is suppressed by the diffusion of the second messengers cGMP

and Ca2+

UAB April 8 2014 – p.34/37

Page 78: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Some Bibliography

• (with P.Bisegna, G.Caruso, D.Andreucci, L. Shen, S. Gurevich and

H.E.Hamm) Diffusion of the Second Messengers in the Cytoplasm Acts

as a Variability Suppressor of the Single Photon Response in Vertebrate

Phototransduction,Biophysical Journal,94(9), (2008), 3363–3384.

• (with P.Bisegna, G.Caruso, D.Andreucci, L. Lenoci, S. Gurevichand

H.E.Hamm) Kinetics of Rhodopsin Deactivation and its Role in

Regulating Recovery and Reproducibility in Rod PhotoresponsePLoS

Comp. Biology6(12), (2010), 1-15.

• (with P.Bisegna, G.Caruso, D.Andreucci, L. Lenoci, S. Gurevichand

H.E.Hamm) Identification of the Key Factors that Reduce the Variability

of the Single Photon Response,PNAS,108(19), 2011, 7804–7807.

UAB April 8 2014 – p.35/37

Page 79: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Acknowledgments• Daniele Andreucci, Univ. of Rome, La Sapienza, Italy

• Paolo Bisegna, Univ of Rome, Tor Vergata, Italy

• Giovanni Caruso, CNR and Univ. of Rome, Tor Vergata, Italy

• Seva Gurevich, Vanderbilt, Pharmacology

• Heidi E. Hamm, Vanderbilt, Pharmacology

• Fred Rieke (HHMI), Physiology and Biophysics, Seattle WA

• Clint Makino, Harvard Med. School, Boston MA

• Ralf Mueller, Vanderbilt, Pharmacology.

• Lixin Shen, Biomathematics Study Group, Vanderbilt.

• Support: NIH-1-RO1-GM 68953-01; NAS-Keck Foundation

• Web site: http:abcd.math.vanderbilt.edu/˜signaltr/

UAB April 8 2014 – p.36/37

Page 80: Visual Transduction: A Paradigm for Signaling Transduction: A Paradigm for Signaling E. DiBenedetto em.diben@vanderbilt.edu Department of Mathematics, Vanderbilt University Nashville,

Vision-VI

• Parameter Determination is an Art of its Own

• However .... Beware of Parameter Fitting

UAB April 8 2014 – p.37/37