Lecture 13
Multiplicity and statistical
definition of entropy
Readings:
Lecture 13, today: Chapter 7: 7.1 – 7.19
Lecture 14, Monday: Chapter 7: 7.20 - end
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Today’s Goals
Concept of entropy from statistical thermodynamics
Bulk properties of large collections of events/molecules
Multiplicity
Macro and microstates
Statistical definition of the second law of thermodynamics
For a spontaneous process, the multiplicity of the system
increases
Statistical treatment of multiplicity explains macroscopic
properties – back to the ideal gas expansion
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Hemoglobin and probability
Oxygen binding molecule. Its quaternary structure is a
tetramer composed of two α- and β- subunits.
Assume that for each heme:
p(bound)=1/2
p(unbound)=1/2
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Independent events
½
½
½
½
The probability of two
independent events is the
product of their individual
probabilities.
p = ½ x ½ x ½ x ½ = 1/16
½
½
½
½
½
½
½
½
Therefore, for each
hemoglobin, the probability of
obtaining any given outcome is
1/16
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Heme position
1 2 3 4
Multiplicity
What if all you care about is the number of
heme groups that either have or do not have
oxygen bound?
Aggregate the relevant possible outcomes
together
# of desired outcomes
p
total # of possible outcomes
Number of desired outcomes is the
multiplicity, W
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Multiplicity of a molecular system
The multiplicity of a molecular system is the number of
different molecular configurations consistent with the
macroscopic parameters (e.g. temperature, volume,
concentration) that define the system
A microstate is one particular configuration of molecules
that is consistent with the global macroscopic parameters
that define a particular state of the system
The multiplicity is the number of microstates
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Heme position
1 2 3 4
Multiplicity
4 bound & 0 unbound → W = 1 → p = 1/16 = 0.0625
3 bound & 1 unbound → W = 4 → p = 4/16 = 0.25
2 bound & 2 unbound → W = 6 → p = 6/16 = 0.375
1 bound & 3 unbound → W = 4 → p = 4/16 = 0.25
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0 bound & 4 unbound → W = 1 → p = 1/16 =0.0625
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How to compute multiplicity
When we get to a large number of outcomes / microstates,
it gets tedious to classify and count all of these
For the binomial case (i.e. two possible outcomes, such as
bound/unbound), we can generalize this as:
M!
W
N!( M  N )!
Where N is the number of bound hemes obtained in a total
of M hemes
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Calculating multiplicity using factorials
Calculate W for 2 bound & 2 unbound in a hemoglobin:
Calculate W for 4 bound& 0 unbound in a hemoglobin:
Note: 0! = 1 (the product of no numbers at all is 1)
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Ways of binding oxygen
Four hemes, labeled 1
through 4
(numbers represented by
different colors)
In how many possible
ways can the hemes
be chosen?
4 choices x
3 choices x
2 choices x
1 choice
4!=24
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Multiplicity of 2 bound & 2 unbound
2
crimson
bound
sites
2 blue
unbound
sites
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Multiplicity of 2 bound & 2 unbound
2 crimson
bound
sites
2 blue
unbound
sites
In the 24 outcomes, there are
several instances of different
outcomes with the same
bound/unbound pattern
We’re over-counting these
instances
By how much?
There are 2! ways of rearranging
the bound sites and 2! ways of
rearranging the unbound sites
W = 4!/(2! x 2!) = 6
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Hemoglobin multiplicity
Outcome
4 bound & no unbound
3 bound & 1 unbound
Example
M!
W
N!( M  N )!
4!
1
4!0!
4!
W
4
3!1!
W
2 bound & 2 unbound
W
4!
6
2!2!
1 bound & 3 unbound
W
4!
4
1!3!
0 bound & 4 unbound
4!
W
1
0!4!
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Over-counting in a bigger dataset
2 hemoglobins, 8 hemes, 5 bound and 3 unbound
Number of ways to pick the 8 sites in random order: 8!
5
3
4
2
1
2
3
1
Two microstates
of the same
outcome
1
1
1
3
4
3
4
2
3
5
2
5
2
Ways to arrange 5 bound states: 5 x 4 x 3 x 2 x 1 = 120 = 5!
Same for the 3 unbound states: 3!
Over-counting for this red-blue pattern: 5! x 3!
Multiplicity = 8!/5!3! = M!/(N!(M-N)!)
Where M = # hemes, N = # bound states and (M-N) = # unbound states
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As M increases…
10 Hemes
Multiplicity (W)
Multiplicity (W)
4 Hemes (1 Hemoglobin)
Bound O2 molecules
Bound O2 molecules
1000 Hemes
Multiplicity (W)
Multiplicity (W)
200 Hemes
Bound O2 molecules
Bound O2 molecules
Modified from The Molecules of Life (© Garland Science 2008)
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250 hemoglobins-1000 hemes
M = 1000, N = 500 oxygen-bound hemes
The multiplicity of this outcome is:
1000!
W
500!500!
Stirling’s approximation is a useful mathematical tool to
calculate values for large factorials
ln n! n ln n  n
Where n is a large number
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Using Stirling’s approximation
M = 1000, N = 500 oxygen-bound hemes
1000!
W
500!500!
Applying a natural log to both sides: ln W  ln 1000!
500!500!
Then:
ln W  ln(1000!)  ln(500!)  ln(500!)
 ln(1000!)  2 ln(500!)
Applying Stirling’s approximation:
ln n! n ln n  n
ln W  1000 ln 1000  1000
 2(500 ln 500  500)
W = e694 ~ 10301
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Probability of N=500?
W500 ~ 10301
Number of possible outcomes: 21000 ~ 10301
p500 = multiplicity / # of possible outcomes
~ 10301/10301 ~ 1
Essentially all sequences of 1000 will contain an equal
number of bound and unbound hemoglobin proteins
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Other outcomes still have (vanishingly small) probability
Important to keep in mind that the probabilities are still a
distribution
Multiplicity (W)
1000 Hemes
Bound O2 molecules
For 1000 hemes (250 hemoglobin molecules),
p(N=0) ~ 1/10301
Figure from The Molecules of Life (© Garland Science 2008)
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Probability and multiplicity
The relative probability of two outcomes is the ratio of
their respective probabilities
Example, 1000 hemes, N=500 vs. N=600
p500 W500 21000 W500


1000
p600 W600 2
W600
It’s also the ratio of their multiplicity!
In our example: ln (W500/W600) = 694 – 674 = 20
W500/W600 = e20 = ~5x108
Again, an equal number of bound and unbound is by far
the most likely outcome
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Maximal multiplicity when the derivative is 0
How can we find the maximal multiplicity?
When the slope or derivative of the function is 0 and
(N ≠ 0) or (N ≠ M)
That is: W = f(N) and dW/dN = 0 is a condition for
maximal multiplicity
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W
W and lnW have the same maximum
So is dlnW/dN = 0
lnW
Bound O2 molecules
Stirling’s approximation
will be extremely helpful
Bound O2 molecules
Modified from The Molecules of Life (© Garland Science 2008)
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Maximal multiplicity
How can we find the maximal multiplicity?
For the binomial case, it can be derived, using Stirling’s
approximation, that:
d ln W
  ln N  ln( M  N )
dN
At a maximum: d lnW/dN = 0, such that
lnN = ln(M-N)
N = M-N
2N = M
M = N/2
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Maximal multiplicity leads to maximal entropy
What we’ve learned from hemoglobin:
The observed outcome of an experiment involving a large number
of trials is the one that corresponds to maximum multiplicity
This is broadly applicable beyond hemoglobin
Hopefully, it’s starting to sound like the 2nd law of
thermodynamics!
Much of thermodynamics is about finding the conditions where
multiplicity is maximal
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Back to molecules: starting with an ideal gas
Isothermal expansion example from Lecture 9
Here dU = dq + dw = 0 because T = constant
Therefore, dq = - dw
Even though dU = 0, there is a change in the system
We stated that “entropy” drives the change
Figure from The Molecules of Life (© Garland Science 2008)
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Back to molecules: starting with an ideal gas
Let’s now consider that this compressed gas is in fact made
of particles, gas molecules
More space is available to these molecules when the gas expands
What happens to the multiplicity of the system?
Figure from The Molecules of Life (© Garland Science 2008)
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Multiplicity in a molecular system
The multiplicity of a molecular system is defined as the
number of different configurations or conformations of the
component particles that are equivalent
Note: in reality, a
configuration includes the
description of kinetic energy
of each molecule. Here, we’ll
focus only on the position
component
Ideal gas: particles have positions in the system volume
Grid boxes of arbitrary size, small enough to uniquely define the
position of the atoms
Figure from The Molecules of Life (© Garland Science 2008)
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Calculating multiplicity
There are 49! ways to put 49 particles in 49 boxes
Figure from The Molecules of Life (© Garland Science 2008)
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6 particles in a box
Similar to the case of bound and unbound hemes:
M = number of boxes
N = number of red particles
(M – N) = number of empty boxes (“blue particles”)
M!
49!
W

 1.4 107
N!( M  N )! 6!(49  6)!
Figure from The Molecules of Life (© Garland Science 2008)
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Multiplicity increases as volume increases
Double the volume of the box and the multiplicity is now:
98!
W
6!(98  6)!
ln W  ln 98! ln 6! ln 92!
 98 ln 98  98  ln 720  (92 ln 92  92)
 20.72
Figure from
The Molecules of Life
(© Garland Science 2008)
W  109
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Atoms are likely to spread
The two configurations shown below are equally likely in a
larger space
But from our calculations, there are ~1.4x107
configurations equivalent to the one on the left, and ~109
configurations to the one on the right
We’re therefore ~100 times more likely to see the atoms
on both sides of the box
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Figure from The Molecules of Life (© Garland Science 2008)
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Equilibrium state of a system
A microstate is a particular configuration of the atoms or
molecules in a system
Instantaneous snapshot
A state is a global description of the system
Uses bulk properties like temperature, pressure, number of
molecules
The multiplicity of a state is the number of corresponding
microstates
The higher the multiplicity, the higher the likelihood of
observing that state
The system will spontaneously evolve towards the state with
highest multiplicity
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Maximal multiplicity defines equilibrium state
Sudden increase in volume
of the system
Over time, the atoms
spread to increase
multiplicity, until they are
evenly distributed, which
has maximal multiplicity
The multiplicity of the
system drives the
expansion
Figure from The Molecules of Life (© Garland Science 2008)
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If we combine systems A and B, what would the
combined W be?
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WA+B is the product of WA and WB
Figure from The Molecules of Life (© Garland Science 2008)
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lnW is an additive and extensive property
System A
Additive
System B
WA
WB
W=WA×WB
ln W = ln WA + ln WB
Extensive
System 1
W=W1
System 2
2x
W=(W1)2
lnW=2ln(W1)
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Use of lnW for large systems
W rapidly becomes unmanageably large, lnW increases
slower with the # of particles
Notice lnW is an extensive and additive property of the
system
Combining systems A and B to yield (A+B)
WA+B = WA x WB
lnWA+B = lnWA + lnWB
lnW is a state function, it depends only on the parameters
of the present state of the system
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Statistical definition of entropy
Entropy is defined as:
S  kB ln W
Where kB is the Boltzmann constant
kB = 1.38 x 10-23 JK-1
= R/NA = 8.314 JK-1/6.023 x 10-23
Entropy is a state function of the system
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Some concepts to remember
Multiplicity (W) is the number of microstates for a given
conformation of the system
A system will spontaneously evolve towards the state of
maximal multiplicity
Entropy is a function of multiplicity
Additive property
Extensive property
State function
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