Energy landscape in the two state model (Boltzmann distribution)

• MikeGG
In summary, the individual probabilities of each energy state can be calculated using the formula e^-Ei/KT, but the challenge lies in incorporating the radius into the equation. One suggestion would be to find a way to relate the radius to the energy states, or to use the Boltzmann Distribution equation to incorporate the radius into the energy landscape plot.
MikeGG
Homework Statement
Draw an energy landscape such as shown in Figure 7.7 for
the voltage-gated sodium channel presented in Figure 7.2.
In particular, show how the landscape changes as a function
of the applied voltages shown in Figure 7.2(C).
Relevant Equations
E(σ ) = σ εopen + (1 − σ )εclosed − στA

Gtension = − τΔs × (R /Rout)× ΔR × (2πRout/s)

Gtension = −τ2πRΔR

ΔΑ= 2πRΔR

Popen = e^−β(εopen−τΔA) /(e^−β(εopen−τΔA) + e^−βεclosed)

β=1/ ΚΤ
From an excel file I can get the probability of each energy state Εi and I saw at Wikipedia that the probability of each energy is proportional with

e^−Εi/KT, from this I find the energy of every micro state. Also from the formula
which I found on a paper I can get a curve like the curve shown in fig7.7 (almost) whith the probabilities I took.

My problem is that it must be an energy as a function of the Radius, so I must put in there somehow the R. I can take the formula with Popen and to put on the denominator all of the energies and on the numerator the energy of each microstate but then I don't have the R.
If i put the whole energies of each state I could put ΔΑ but i don't see how with that I could made curves like fig7.7 and also instead of τ I must put v (volts), the probabilities I took was from -105 mVolts, if I put that the result doesn't makes sense.

So I want the different probabilities to use the formula
but I can't find a way to put radius in. That's the way I am thinking it, I would appreciated if someone could tell me if I am in the right direction and maybe give a hint how to move forward or to give me a hint for another direction.
Also, fig 7.2 has the Δε which indicates the it may be useful for the drawing of the energy landscape but I can't figure out how.

Thanks

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in advance!It seems like you have a good start, but I'm not sure if this approach will work for what you're trying to do. One suggestion might be to try and find a way to relate the radius to the energy states. For example, if you know the size of each energy state and the radius, you could then use that information to calculate the energy of each microstate. You could then use this information to construct your energy landscape, plotting energy against radius. Another suggestion would be to look up the Boltzmann Distribution equation, which gives the probability of occupancy of an energy state as a function of temperature. This equation should also let you incorporate the radius into your plot. Hope this helps!

1. What is the two state model in the context of energy landscapes?

The two state model is a simplified representation of a system with two distinct energy states. It is often used in the study of molecular systems, where the two states represent different conformations or configurations of the molecules.

2. How does the Boltzmann distribution relate to energy landscapes in the two state model?

The Boltzmann distribution describes the probability of a system being in a particular energy state at a given temperature. In the two state model, the energy landscape is determined by the relative energies of the two states and the temperature of the system.

3. What factors influence the shape of the energy landscape in the two state model?

The shape of the energy landscape in the two state model is primarily influenced by the energy difference between the two states and the temperature of the system. Other factors such as external forces or interactions can also affect the landscape.

4. How does the energy landscape affect the behavior of the system in the two state model?

The energy landscape determines the stability and dynamics of the system. The lower energy state is typically more stable and will be more populated at lower temperatures, while the higher energy state may become more populated at higher temperatures or under certain conditions.

5. How is the energy landscape in the two state model experimentally determined?

The energy landscape can be experimentally determined through techniques such as spectroscopy or thermodynamics measurements. These methods can provide information about the relative energies of the two states and their populations at different temperatures.

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