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Problem Optical tweezers permit measurements of
the forces applied to single biological
molecules. Calculate the maximum tensile forces
that a polypeptide chain can withstand, assuming
that the covalent bonds, which hold the backbone
together, can be described by the Lennard-Jones
potential function.
The values for e and r0 for the carbon-carbon
bonds are respectively 0.56x10-18J and
0.152x10-9m, while the corresponding values for
carbon-nitrogen bonds are respectively
0.51x10-18J and 0.149x10-9m.
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Lennard-Jones Potential
r0 distance parameter
e energy parameter
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Course Outline
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Brownian motion, Lagevins equation and the
movement of biomolecules
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Brownian motion
The term Brownian motion derives its name from
the botanist Robert Brown who, in 1828, made
careful observations on the tiny pollen grains of
a plant under a microscope.
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Einstein (1905) first provided a sound
theoretical analysis of the Brownian motion on
the basis of the random walk problem and
thereby established a far-reaching relationship
between

• the irreversible nature of this phenomenon (as
  well as of the phenomena of diffusion and
  viscosity)

• the mechanism of molecular fluctuations.

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Revision of the simple 1D random walk

• x(t) is position of Brownian particle at t.
• Starting point at t 0 is x 0.
• Molecular impact causes particle to jump to the
  left or right a distance l with equal
  probability, so ?xl and ?x-l .
• Impacts take place (on average) after a time ?

Dx
0
Probability to find particle at x after a series
of n(t/?) successive jumps ?
This is the probability that the particle makes
m(x/l) more jumps in the positive direction of
the axis than in the negative direction
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Revision of the simple 1D random walk
Probability is given by the binomial expression
Thus, for tgtgt? we have for the net displacement
of the particle
Hence, the root-mean-square (rms) displacement of
the particle is proportional to the
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Revision of the simple 1D random walk
To obtain an asymptotic form of the probability
function pn(m), we apply Stirlings formula,
to the factorials
and we get
Taking x to be a continues variable we get the
Gaussian form
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Revision of the simple 1D random walk
where
The quantity D introduced here is identical with
macroscopic diffusion coefficient of the given
system.
The Einstein-Smoluchowski equation relates the
diffusion constant D and the friction coefficient
b
b depends on the size and the shape of the
diffusing particle
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Diffusion equation
The number density of the Brownian particles in
the fluid is and the
current density by the symbol then according
to Ficks law,
The equation of continuum is
Substituting we obtain the diffusion equation
(Ficks 2nd law)
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Diffusion equation
With the solution
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Diffusion equation
Using different units (dividing by NA ) the
diffusion equation can be rewritten
This is a partial differential equation, its
solution yields c(r,t). To solve it one requires

• Two boundary values, concentration at particular
  points in space

• One initial condition, concentration
  distribution at some particular time

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Example
Protein with smaller molecules (e.g. ATP) or
Ribosome with tRNA
Calculate the rate of collision of small
particles with A in steady state under the
assumption that c remains constant far away from
A and that the particles are absorbed, dissipated
or transformed once they collide with A.
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Example
To compute the rate, solve the diffusion equation
in spherical coordinates.
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Example
1st Integration
2nd Integration
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Example
Number of collisions per second at r r0, called
the current I(r0)
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Langevin equation and autocorrelation functions
is the friction coefficient
F(t) is a random force
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Particle in fluid
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The time autocorrelation function
can be used to analyse the memory of a system

• Similar correlation functions can be obtained
  for position r or forces F

• Indicates how fast a particle forgets its
  initial velocity, owing to Brownian random motion

• for very short time t after 0 the correlation
  function has the value

• for long time t after 0 the velocity is
  completely uncorrelated so

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Velocity autocorrelation function from the
Langevin model
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Velocity autocorrelation function from the
Langevin model
Integration from t 0 to infinity
Green-Kubo relationship