Question

Biochemical Statistical Thermodynamics: Model vs. Scientific Evidence

Answer 1

I have a minor problem I like to discuss with all you brilliant people:
I have been trying to model a model a simple hexapeptide in order to look upon the probabilities for the protein to denature under increasing temperature.
The model I have been using is shown in attachment 1.
I have choosen to ignore any translational, rotational and vibrational contributions. The energy for the state is given to:
\[N_{A} \epsilon _{0}=2.5 kJ/mol\]
Doing this I have been writing the equation for the partition function:
\[\LARGE q=\sum_{j}^{states}e ^{- \beta \epsilon _{0}}=\sum_{i}^{levels} g _{i} e ^{- \beta \epsilon _{i}}\]
\[\LARGE \beta= \frac{ 1 }{ kT }\]
So for the system it most be:
\[\LARGE q=4e ^{-\frac{ 0 }{ kT }}+11e ^{-\frac{ \epsilon _{0} }{ kT }}+21e ^{-\frac{ \epsilon _{1} }{ kT }}=4+11e ^{-\frac{ \epsilon _{0} }{ kT }}+21e^{-\frac{2 \epsilon _{0} }{ kT }}\]
The possibilities using the Boltzmann distribution are given by:
\[\LARGE p _{i}=\frac{ e ^{- \beta \epsilon _{i}} }{ q }\]
Doing so we can 3 possibility vs. temperature plots for the 3 states and can be found in attachment 2. (pre-calculation found in attachment 3)

We can from the plot observe that as the temperature in the possibility for the hexapeptide to get denatured increases. This is what we expect from general observations and theory. but model also suggest that at all temperatures there are a chance to become denatured, i is small but it is there. This is what the question is about:
Does scientific evidence support that suggestion, or is one of the weakness of the model.

Usually when we look upon reaction rate under increasing temperature (below the denaturation level) we see the reaction rate increase, this alone does not support the models suggestion, but this may be because the temperature bellow denaturation level have a higher contribution to the reaction rate than the possibility of getting denatured. But at some point we get to a critical level where the possibility become higher than the contribution to the reaction rate and the rate decrease due to
\[\LARGE v \propto [E]\]

I have been calculating the mean nr. of hydrogen bonds using that:
\[\LARGE \langle n_{c} \rangle=\sum_{i}^{states} n_{c} p _{i}\]
Doing so I get:
\[\LARGE \langle n_{c _{{T=300K} }} \rangle=1.1076\]
So the model suggest that there should average be 1.11 hydrogen bonds pr. molecules.

Answer 2

@Preetha @blues @ataly

Answer 3

@chmvijay
@nincompoop
@abb0t

Answer 4

Unfortunately, I haven't studied this one on molecular-level for me to provide an insight to the model you posited. Though the cited observations on increased in reaction rate relative to temperature you mention are found in many literature. One of the things that I have been studying this week was Δ↑ in temperature results in ↑ in production of enzymes in living systems particularly the closed-circuit ones. My reluctance on positing theoretical models without basing from observations stems from emergent properties. The attributes and nature of molecular structures change with regards to increasing or decreasing complexities. @abbot however, might give you a better insight since he has spent more time learning this than I have.
THOUGH I THINK FOR SURE THAT YOU ARE IN THE RIGHT TRACK.

Answer 5

I haven't studied this intensely on molecular-level requiring mathematical models like the one you made or outside Vander Waal's. Organic chemistry is a little bit more complicated due to the multi-interactions going on within a living-system like the feedback mechanisms in homeostasis. Again, sorry that I couldn't offer any help.

Answer 6

It did not write the full partition function:
\[\large q=4e ^{-\frac{ 0 }{ kT }}+11e ^{-\frac{ \epsilon _{0} }{ kT }}+21e ^{-\frac{ \epsilon _{1} }{ kT }}=4+11e ^{-\frac{ \epsilon _{0} }{ kT }}+21e^{-\frac{2 \epsilon _{0} }{ kT }}\]

Answer 7

-_-