# Boltzman distribution

John Misasi (jmisasi@engc.bu.edu)
Tue, 9 Apr 1996 03:47:20 -0400 (EDT)

Hi everyone=85 If you don=92t remember me my name is John Misasi and I am a
Biomedical engineering senior at BU. I promised I would post after I
returned from my spring break, over a month ago, and this is my attempt!

INTRODUCTION:

I wish to discuss the Boltzman=92s distribution function and how it may=
relate
to the origin of a biomolecule in some sort of pre-Earth environment. This
probably is related to the "Is it soup, yet" thread. I come into this
discussion with a interest in this question, but I feel it necessary to
declare a bias of not being a believer in the pre-biotic soup. Still, I
wanted to know what Temperatures (T) were needed in order to support
different ratios of amino acid isomers (i.e. the ratio of D:L ).

For those who don=92t know or do not recall the Boltzman distribution
function, it can be used to relate the ratio two states to energy and
temperature via the following formula:

Ni
(1) ---- =3D A * exp[ -(Ei-Ej) / kT ]
Nj

where Ni and Nj are the two states, A is a constant that is dependent on the
system, E is the energy in either state i or j, k is the Boltzman constant (
1.3807 X 10 (-23) J / K ) and T is temperature in Kelvin.

This can be used to determine the ratio of a high energy molecule to a low
energy molecule at a given temperature. The molecules I would like to
discuss using this distribution is the distribution of amino acids between
the L and D forms. I realize that current theory hypothesizes an RNA world
first, but I haven=92t thought enough about how to apply Boltzman to that
environment. =20

It is well known that most of the amino acids found in living systems are of
the L form. This is not because L is favored thermodynamically, it just
seems that life favors L over D and has incorporated L stereo-specificity
into the machinery. The question arises, Why L not D? Also, how likely is
it that L amino acids are favored over D amino acids ? This latter question
can be answered by a proper application of the Boltzman distribution
function, which I will attempt ;^) to apply.

The change in energy of this system can be calculated by assuming that the
only energy input into the system, other than T, will be electromagnetic
energy. Therefore, one can calculate the electromagnetic energy by using
the formula:

(2) Ei - Ef =3D h * f where h is Planck=92s constant =3D 6.6261 X
10(-34) J s
and f is frequency in Hertz.

Lastly, in order to calculate temperature we must manipulate of Equation 1
to the following form:

( Ei - Ej )
(3) T =3D ----------------
k*ln(Ni / Nj)

Methods:

For simplicity, the constant A will be set equal to 1.

There are two states which can be arbitrarily assigned to Ni or Nj because
neither states is more energetically favored. I will assign Ni to D and Nj
to L. I chose this because it allows a ratio less than 1 which is the
maximum probability.

I also chose to use high energy wavelengths of light for equation (2):200,
300, and 400nm.

I simply calculated the various temperature using equation (3) at varying
ratios of isomers and the energy at the different wavelengths of light.

Results:

Table 1: Boltzman Distribution Temperature Calculations=97sample of=
collected
Boltzman temperature.

D=3DL (Racemic mixture) Ni:Nj =3D 1 T =3D infinity

L>>D Ni:Nj =3D> infinity T =3D 0 K

D>>L Ni:Nj =3D> 0 T =3D 0 K

Ni / Nj T (200 nm) T (300 nm) T (400 m)
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D

7/8 539 E3 359 E3 269 E3
2/3 177 E3 118 E3 88.7 E3
0.5 103 E3 69 E3 51.9 E3
E-2 15.6 E3 10.4 E3 7.81 E3
E-5 6.25 E3 4.17 E3 3.12 E3
E-10 3.12 E3 2.08 E3 1.95 E3
E-50 625 E0 417 E0 312 E0
E-99 316 E0 210 E0 158 E0

Note: If the ratio is inverted, resulting in a ratio larger than 1, the
answers would be equal and opposite in value.

Figure 1: Graph of Boltzman Data-- this can be seen at:=20
http://eng.bu.edu/CAB/jmisasi/Boltzman.jpg

The data in Table 1 and Figure 1 show that as the ratio of D:L decreases the
Temperature required decreases. Temperatures drop from a maximum of 539 000
K to a minimum of 158 K. If this graph were extrapolated to 1 (i.e. a
racemic mixture) the temperature needed would be infinite and a mixture of
all L isomer leads to a temperature at absolute zero. Increased wavelength
or decreased energy causes the system to decrease in temperature at a given
isomeric ratio.

Discussion:

Figure 1 shows that it is extremely unlikely that we would ever see a purely
racemic mixture of one isomeric form, because of the infinitely high
temperature that is needed. Similarly, since the Earth is rarely at
absolute zero, we should not expect to see only one isomeric form.
Next, not knowing the average temperature of the water on Earth 3-4 billion
years ago ( any help would be appreciated), I would assume a temperature
between 273 K (freezing) and 373 K (boiling). This puts the concentration
ratio seen in Table 1 between E-50 and E-99. This is a huge ratio. One
isomer is going to dominate, either D or L. Recall, I can easily switch D
with L and get the same result. Therefore, the question remains: Which one
will dominate and Why? This seems to be only a matter of chance in a
pre-biotic soup! =20
Empirically, if this is the case, equal probability of either dominating,
it would seem whichever form was made first would be the dominating form.
But I would bet there are a few chemists out there who would dispute that
because if they form a mixture of amino acids they get a racemic mixture!
This I cannot explain. Anyone have a thought or answer ?
Lastly, as I said in the Intro. I wonder how this type of probability
function might be applied to the new hypothesis of a pre-biotic RNA world.
What might the two states be: ? Could it be useful RNA molecule vs. useless
RNA molecule ? If one knows the energy of the system and an approximate
temperature, one could determine the distribution of useful RNA molecules to
useless RNA molecules. What do you guys think? =20

Sorry about the length of this but I wanted to give a coherent analysis.

John Misasi
____________________________________________________________________________
John Misasi
jmisasi@bu.edu
@engc.bu.edu
@darwin.bu.edu