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Thermo: A brief Intro
I. Microstates and Global States
II. Probability
III. Equilibrium
IV. Free energy surfaces
Microstate
Microstate: Specify the molecule are uniquely as possible
In a quantum description this could include:
Electronic state
Vibrational and rotational quantum numbers
Spin states
Position of all atoms
We do CLASSICAL descriptions
positions of all atoms defines a microstate
(defer discussion of surroundings)
Global
Global: A (user-defined) sum over a set of microstates
Often a sum over sets that cannot be distinguished somehow:
due to experimental limitations
Due to similarity of structure
Due to similarity of function
Common uses:
Folded vs. unfolded
Folded vs. unfolded vs. intermediate
different functional states
Population: Microstate
When we have multiple states, they appear in different populations depending on
their energies:
 (  j 1 )
Pj 
nj
N

 je
k BT
 e
i
 (  i 1 )
k BT
i
Each microstate has its own population.
N. B. The bottom is the canonical partition function.
Assuming near equilibrium and a large system
Notes: This can be proven; those of you in physics/physical chemistry probably
will do so or have done so already. This is also the high T limit of the fermi-dirac and
Bose-Einstein distributions
Population: Global
When we have multiple states, they appear in different populations depending on
their energies:
 ( j 1 )
Pj 
nj
N
 je

k BT
 e
i
 (  j 1 )
i
 ( i 1 )
k BT
Pj 
nj
N

 je
k BT
 e
i
 (  i 1 )
k BT
i
Each global state has its own sum over microstates. This is sometimes represented by,
 (  j 1 )
Pj 
nj
N

 je
k BT
 e
i
i
 ( i 1 )
k BT
Where j is now considered as a global state
energy.
Probability and the
partition function
When we have multiple states, they appear in different populations depending on
their energies:
 (  j 1 )
Pj 
nj
N

 je
k BT
 e
i
 (  i 1 )
k BT
i
The bottom is the canonical partition function, Q.
Equilibrium
We assume equilibrium,
What is equilibrium?
Does equilibrium mean stasis?
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0
0
-1000
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-10000
-3000
{pressure and temperature over ~200ps in ~130,000 atoms system}
Fluctuations
Fluctuations (in macroscopic quantities) occur during equilibrium
The fluctuation-dissapation thm says that we can perturb a structure, and
that relaxation from that perturbation is equivalent to relaxation from the
same fluctuation
200
150
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0
-50
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-150
-200
Different physical properties fluctuate differently
{pressure and temperature over 10ns in ~130,000 atoms system}
Fluctuations
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Larger systems: lesser fluctuations:
~16,000 atoms vs. ~130,000
(Gibb’s) Free Energy
The most fundamental quantity in biological stat mech: Different reps:
DG=-KBTN ln Q
DG=-RT ln K
G=H-TS -> DG=DH-TDS
Free Energy Surface
Free energy as a function of something
Tells us about minima, and saddle point (transition points)
In principle, given a free energy surface we can deduce the behavior of
the system over time
Erogodic principle: Given sufficient times, all microstates will be sampled regardless
of initial conditions
Proteins generally are regarded as having special free energy surfaces
Free Energy Surface
Proteins generally are regarded as having special free energy surfaces
folding funnel with a well-defined global minima
This “minima” is a global state that may consist of substates that are
well-connected, or not. Still an open question
Flexibility
Experiment: the thermodynamics of HEWL and a mutant missing a disulfide bond
were studied with scanning microcalorimetry.
Both proteins have the same enthaply of unfolding, and had “two-state” behavior, but
there is a difference in the entropy of unfolding.
The x-ray structures show essentially the same structures and interactions.
Where does the entropy change come from?