Post on 08-Jun-2018
Visual Transduction: A Paradigmfor Signaling
E. DiBenedettoem.diben@vanderbilt.edu
Department of Mathematics, Vanderbilt University
Nashville, TN 37240, USA
UAB April 8 2014 – p.1/37
Micrograph of Salamander and Frog Retina
UAB April 8 2014 – p.2/37
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
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
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
.......................................................................................................................................................................................................................................................................................................................... ................................................................ ........................................................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .............................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .............. ...................................................................................................
............................................................................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ...............
.....................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................ .............. ............................................................................................
......................................................................................................................................................................................................................................................................................................................................................
............................................................................................................... ......................................................................... ......................................................................... .........................................................................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
Incisures in Frog ROS
Vision Res. Vol. 22, 1982, pp. 1417–1428
UAB April 8 2014 – p.6/37
Incisures in Rat ROS
Vision Res. Vol. 7, 1967, pp. 829–836UAB April 8 2014 – p.7/37
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
Light Activation
UAB April 8 2014 – p.9/37
The Signaling Cascade on the Activated Disc
UAB April 8 2014 – p.10/37
The Cascade and Resulting Photocurrent
UAB April 8 2014 – p.11/37
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
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
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
Vision-II
• Combining Biophysics, Biochemistry and
Geometry/Anatomy
• (subtitle) System Behavior
UAB April 8 2014 – p.12/37
Revisiting the Process by Injecting 3 New Ideas
UAB April 8 2014 – p.13/37
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
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
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
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
.......................................................................................................................................................................................................................................................................................................................... ................................................................ ........................................................................................................................................................................................................................................................................................................................................................ ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .................................................................................................................................................................................... .............................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................................................. .............. ...................................................................................................
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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.14/37
The Formal Limiting Domains-I
Vε −→ V = ro < x1 < R × x2 = 0DR+σε − Vε −→ DR − VDR+σε − DR −→ ∂DR
...................................................................................................................................
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...................................................................................................................................................................................... ................................................................................................................................................................................................................. ................................................................................................................................................................................................................. roro O O
x1 x1x2x2
UAB April 8 2014 – p.15/37
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)
....................................................................................................................................................................
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.......................................................................................................................zz
x1x1 x2x2
UAB April 8 2014 – p.16/37
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
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
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
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
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
Vision-III
• Novel Mathematics
• (subtitle) Partial Differential Equations in
Fractured Domains or Domains with Spikes
UAB April 8 2014 – p.19/37
The form of f(u) Contains the Biochemistry
UAB April 8 2014 – p.20/37
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
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
Simulated and Experimental Current Drop 1 − J/jdark
UAB April 8 2014 – p.21/37
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
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
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
Single Photon. Current as a Function of the Incisures
(Salamander)
UAB April 8 2014 – p.22/37
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
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
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
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
Vision-IV
• Numerical Analysis, Programming,
Simulating
• (subtitle) Capability of Replacing Wet-Bench
Experiments with Virtual Experiments
UAB April 8 2014 – p.23/37
The Activation/Deactivation Cascade, Variability and
Fidelity of the SPR
UAB April 8 2014 – p.24/37
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
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
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
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
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
The Activation Cascade and Quenching
UAB April 8 2014 – p.25/37
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
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
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
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
Variability Versus Biochemistry
UAB April 8 2014 – p.26/37
Variability Versus Biochemistry
• If CV ≈ 1/√
n (Poisson statistics), then one needs about20
Biochemical steps to quenching
UAB April 8 2014 – p.26/37
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
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
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
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
Choosing a Biochemistry
UAB April 8 2014 – p.28/37
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
Continuous Time Markov Chain (CTMC):
Variability and Biochemistry
UAB April 8 2014 – p.29/37
Continuous Time Markov Chain (CTMC):
Variability and Biochemistry
UAB April 8 2014 – p.29/37
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
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
Simulations and Functionals
UAB April 8 2014 – p.30/37
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
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
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
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
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
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
Conclusions
• Homogenization and Concentrated Capacity permits one to extract
functional properties of the cell architecture. In particular:
UAB April 8 2014 – p.34/37
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
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
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
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
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
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.
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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/
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Vision-VI
• Parameter Determination is an Art of its Own
• However .... Beware of Parameter Fitting
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