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Outline Physiology Overview Hypothesis Mathematical Toolbox Conclusion

The Big SqueezeWhy strain is so exciting to myocardium

Geoffrey A.M. Hunter

Department of MathematicsUniversity of Utah

IGTC Graduate Research Summit 2007

September 29, 2007

Geoffrey A.M. Hunter: IGTC Graduate Research Summit 2007

Outline

Physiology overview:Electrophysiological changes caused by strain.How does regional ischemia lead to localized strain?Identifying the protein subunit of the human stretch-activated channeland its structure.

Hypothesis

Mathematical toolbox:Hidden Markov model of a stretch-activated channelGillespie algorithm for single-channel simulationsMaximum Likelihood optimization for finding an optimal hidden Markovmodel

Future work & sideline interests

Outline

Hypothesis

Outline

Hypothesis

Outline

Hypothesis

Electrophysiological Changes from Strain

It’s well known that an electrical stimulus causes a mechanical response in the heart(contraction), but mechanical stimuli can generate electrical responses as well.

The Good: Strain can explain the Frank-Starling Law whereby increaseddiastolic volume leads to increased systolic contraction, which dynamicallymaintains proper blood flow in the heart (i.e. rate in = rate out).

The Bad: A blunt impact to the chest during the refractory period of the actionpotential can kill a healthy athlete immediately (Commodio Cordis).

Changes to the action potential:

Figure courtesy of Dr. Frank B. Sachse

Strain in the Border Zone

We tend to think of the genesis of localized strain as follows...

During regional ischemia, ischemic myocardium is known to contract slower andgenerate less force than normal myocardium.

Because of this inhomogeneity, the ischemic region bulges outward as it’s notable to support the contractile load on the heart like the normal region is able to.

The border zone (see below) is an area where large gradients in strain are foundand is also where excitatory currents are generated.

Some channels, called stretch-activated channels, are activated by strain andthe protein subunits of these channels were recently identified.

Identifying TRPC1

Maroto et al. 2005 used purified oocytemembrane to identify TRPC1 as the proteinsubunit of the human non-selectivestretch-activated channel.

Properties:

Small conductance∼ 0mV reversal potentialPermeable to cations onlyStrain in the cell membrane was sufficientto activate TRPC1Weak voltage dependenceForm homotetrameric channels

(Maroto et al. 2005)

Identifying TRPC1

Properties:

Small conductance

∼ 0mV reversal potentialPermeable to cations onlyStrain in the cell membrane was sufficientto activate TRPC1Weak voltage dependenceForm homotetrameric channels

Identifying TRPC1

Properties:

Small conductance∼ 0mV reversal potential

Permeable to cations onlyStrain in the cell membrane was sufficientto activate TRPC1Weak voltage dependenceForm homotetrameric channels

Identifying TRPC1

Properties:

Small conductance∼ 0mV reversal potentialPermeable to cations only

Strain in the cell membrane was sufficientto activate TRPC1Weak voltage dependenceForm homotetrameric channels

Identifying TRPC1

Properties:

Small conductance∼ 0mV reversal potentialPermeable to cations onlyStrain in the cell membrane was sufficientto activate TRPC1

Weak voltage dependenceForm homotetrameric channels

Identifying TRPC1

Properties:

Small conductance∼ 0mV reversal potentialPermeable to cations onlyStrain in the cell membrane was sufficientto activate TRPC1Weak voltage dependence

Form homotetrameric channels

Identifying TRPC1

Properties:

Small conductance∼ 0mV reversal potentialPermeable to cations onlyStrain in the cell membrane was sufficientto activate TRPC1Weak voltage dependenceForm homotetrameric channels

Structure of TRPC1 Proteins & Channels

6 transmembrane spanning subunits with S5-S6 forming the pore

Known to associate with structural proteins (Cav-1, Homer-1, Ankyrin),signalling molecules (Calmodulin), and other channels (IP3RI & IP3RII) but therole of these relationships in myocardium is unknown.

Permeability ratios (PNa : PK : PCa): 1:0.95:0.23

Hypothesis: Role of stretch-activated channelsWe suggest that the biphasic changes in conduction velocity can be explainedby the activation of stretch-activated channels.

Hypothesis: Role of stretch-activated channels

We suggest that the biphasic changes in conduction velocity can be explainedby the activation of stretch-activated channels.

Hypothesis: Structure of border zone

The microscopic structure of the border zone is unknown (i.e. width, levels of ATPand electrolytes, distribution of normal and ischemic cells), so we want to look atdifferent profiles of these factors in the border zone to see if there are significantdifferences in action potential morphology, conduction velocity, and excitability.

Mathematical Tools

Our modelling work has utilized the following tools so far:

1 Hidden Markov model to model the stochastic behavior of a singlestretch-activated channel

2 Gillespie algorithm to make statistically and numerically accuratesimulations of single-channel behavior

3 Maximum Likelihood optimization to find the “best fit” model (i.e.parameters and model topology)

Hidden Markov Model

TRPC1 forms homotetrameric channels that are activated by strain

α0−→←−β1

α1−→←−β2

α2−→←−β3

α3−→←−β4

k+−→←−k−

where βi+1 = (i + 1)b0, αi = (4− i)a(λ) for i = 0, ..., 3 with b0, k−, and k+

constant and a(λ) is a function of membrane strain, λ.

This can be written in matrix-vector form as:

dt= A x

−α0 β1

α0 −(α1 + β1). . .

. . .. . .

. . .. . . k−k+ −k−

Hidden Markov Model

TRPC1 forms homotetrameric channels that are activated by strain

α0−→←−β1

α1−→←−β2

α2−→←−β3

α3−→←−β4

k+−→←−k−

where βi+1 = (i + 1)b0, αi = (4− i)a(λ) for i = 0, ..., 3 with b0, k−, and k+

constant and a(λ) is a function of membrane strain, λ.

This can be written in matrix-vector form as:

dt= A x

−α0 β1

α0 −(α1 + β1). . .

. . .. . .

. . .. . . k−k+ −k−

What’s “hidden” about it anyway?

Definition (Hidden Markov Model)

“A Markov model where the observation value is a probabilistic function of state.These models have a doubly stochastic process where the underlying stochasticprocess that is not observable can only be observed through another set of stochasticprocesses that produce the sequence of observations.” (Rabiner 1989)

This means that...

“...the underlying stochastic process that is not observable...”:

“...set of stochastic processes that produce the sequence of observations.”:

The likelihood of a model, θ, generating a sequence of observations is defined as:

Lik( θ) ≡n∏

P(Oi | θ)

where P(Oi | θ) = the probability observing Oi at time ti given the model θ.

This means that...

Lik( θ) ≡n∏

P(Oi | θ)

This means that...

Lik( θ) ≡n∏

P(Oi | θ)

This means that...

Lik( θ) ≡n∏

P(Oi | θ)

The Gillespie Algorithm

Numerical method for generating numerically and statistically accuratesingle-channel data.

Let aµ(t) dt be the propensity of the µ’th reaction, Rµ (i.e. a2(t) = α1(λ)C1).You can show that the probability that the next reaction will occur in theinfinitesimal time interval (t + τ, t + τ + dτ) given that the system is in stateY = (C0, C1, . . . , C4, O) at time t is:

P(τ, µ|Y, t) dτ = aµ(t + τ) e−∫ τ0 aµ(t+τ ′) dτ ′ dτ

→ F (T , µ|Y, t) =

0P(τ, µ|Y, t) dτ

= 1− e−∫ T0 aµ(t+τ ′) dτ ′

For each reaction, we set u = F (T , µ|Y, t) where u ∼ U(0, 1) and integrateeach equation until: ∫ T+t

taµ̂(τ) dτ = − ln(1− u)

at time T̂ for some µ̂. We update the new time to t + T̂ and update the systemaccording to reaction µ̂ (i.e. if µ̂ = 2, then C1 → C1 − 1 and C2 → C2 + 1).

→ F (T , µ|Y, t) =

0P(τ, µ|Y, t) dτ

= 1− e−∫ T0 aµ(t+τ ′) dτ ′

taµ̂(τ) dτ = − ln(1− u)

→ F (T , µ|Y, t) =

0P(τ, µ|Y, t) dτ

= 1− e−∫ T0 aµ(t+τ ′) dτ ′

taµ̂(τ) dτ = − ln(1− u)

→ F (T , µ|Y, t) =

0P(τ, µ|Y, t) dτ

= 1− e−∫ T0 aµ(t+τ ′) dτ ′

taµ̂(τ) dτ = − ln(1− u)

Why use the Gillespie algorithm?

The random behavior of the channel is preserved with the Gillespie algorithm, whilethe Master Equation produces the expected behavior.

How do you find the “best fit” model?

We have to redefine our notion of “best fit” because the channel recordingdata is...

...noisy from seal resistance, the electronic amplifier, and from Brownian motionof the channel and the membrane.

...Markovian, so data will always be unique even in the absence of noise.

...lost when sampled at a low sampling frequency and when it’s quantized.

...sometimes non-stationary.

Maximum Likelihood Optimization

Can define “best fit” in the following two ways:

1 A model that is most likely to reproduce the statistics of the data (i.e.distributions of dwell times and conductances).

2 A model that is most likely to reproduce the sequence of data measured.

Lik( θ) ≡n∏

P(Oi | θ) →d ln(Lik( θ))

d θ= 0

(Note: There are also other methods to define “best fit”)

Lik( θ) ≡n∏

d θ= 0

Lik( θ) ≡n∏

d θ= 0

Lik( θ) ≡n∏

d θ= 0

Lik( θ) ≡n∏

d θ= 0

Future Work

Currently running Maximum Likelihood optimization on the model.

Hope to find a functional relationship between strain and αi (i.e Is αi (λ) ∝ λ oris αi (λ) ∝ eηλ).

Later we will integrate the TRPC1 model into single cell, 1-D chain, and 2-Dmyocardium models then look at changes in action potential morphology,conduction velocity, etc. in normal and ischemic myocardium.

Test different border zone profiles to see if there are any differences on themacroscopic level

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Special Thanks To...

Dr. James Keener ([email protected]): Supervisor

Dr. Frank B. Sachse ([email protected]): Thesis committee

Dr. Owen P. Hamill ([email protected]): hTRPC1 data

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