Transport
• There are only three ways that particles (such as
molecules or organisms) can move from place to
place:
l
(1) Convection or advection: mass movement.
• E.g.,
E g fluid flow
(2) Active transport: with expenditure of chemical
energy.
• E.g., across cell membranes via channel proteins:
– Trans-membrane proteins found in the phospholipid
bilayer membranes of living cells.
• Movement of organisms under their own power.
Transport
(3) Diffusion:
Diff i
random
d
movement,
t e.g. including:
i l di
• Brownian motion:
– Biologist
o og st Robert
obe t Brown
o
first
st wrote
ote in tthe
e 19
9th ce
century
tu y
about the random motion of pollen grains in a dish of
water.
– Theory
y developed
p by
y Albert Einstein and others.
• Osmosis:
– Net movement of solvent molecules through a partially
permeable membrane into a region of higher solute
concentration.
• Random walk:
– Mathematical formalization
f
off a path that consists off a
succession of random steps.
– Brownian motion: unbiased random walk.
– Osmosis: biased random walk.
Random walks
• Examples of random walks:
– Brownian motion: unbiased random walk.
– Osmosis: biased random walk.
walk
– Genetic drift: unbiased random walk through a state
space.
– Natural selection: biased random walk through a state
space.
Diffusion
• Diffusion model is used to describe random
movement of many types of particles:
– Molecules
– Small aquatic organisms
– Larger animals
• Plays
Pl
iimportant
t t role
l iin many lif
life processes:
–
–
–
–
Directed movements along concentration gradients.
Gas exchange across respiratory surfaces.
Fluid exchange across excretory surfaces.
Searching behavior of organisms.
• Usually effective over small distances.
– Rate is proportional to concentration
difference between two regions.
Random walk in one dimension
• Suppose that we
we’re
re studying the random movement of a
fish in a narrow stream (1D random swim).
– Suppose
pp
that an individual fish moves at random to right
g
or left once every second (unbiased).
• Jumps are of fixed step-length, s.
• Initially assume that fish
f
always moves to the left
f with
probability 0.5, or to the right with probability 0.5.
• Consecutive movements are independent of one another.
• Binomial.
– Possible questions:
• What course would the fish
f
be likely to take?
?
• What would be the distribution of its position after N
seconds?
• How do ‘decisions’ that individuals make translate into
patterns of population abundance at different locations?
Random walk in one dimension
• Some instances (instantiations) of random walks:
p( L)  0.5
Time
p( R)  0.5
05
Displacement
p
p( L)  0.8
p( R)  0.2
Time
Time
p( L)  0.6
p( R)  0.4
Displacement
Displacement
Random walk in one dimension
• Random walks over many realizations (instantiations):
– Can be modeled by a binomial distribution.
– Can be approximated
pp
byy a normal distribution.
p( L)  0.5
p( R)  0.5
– Mean?
– Variance (=standard deviation2)?
Random walk in one dimension
• Consider the number of right moves, R, in N
seconds (steps), where p=Pr(R) is the probability of
a rightward move.
move
• Equivalent to distribution of the number of
‘successes’ in N trials, where p
probability
y of success
is p.
– Binomial distribution: B(N, p).
• X = position of fish relative to starting point:
= Difference between numbers of rightward and
leftward steps
steps,
– Times the size of the step, s:
X   R   N  R   s   2 R  N  s
Random walk in one dimension
• Thus the distribution of position after N steps is a
scaled binomial (multiplied by 2s), with mean
displaced.
p
• Can estimate the mean position and the variance of
position as time increases.
• For binomial distribution B(N, p):
– Mean:
E [ ] = ‘expected value’
E  R   Np
(  p)
– Variance: var  R   Npq  Np (1
• Substituting t for N to get time, we obtain:
E  X    2 E  R   N  s   2 Np  N  s  Ns  2 p  1
 ts  2 p  1
var  X   4 s 2 var  R   4 s 2 Np 1  p 
 4 s 2tp 1  p 
s  step length
s 2  square of step length
Random walk in one dimension
• So, mean and variance of position change with time:
E  X   ts  2 p  1
var  X   4 s 2tp 1  p 
• If p = 0.5,
0 5 simplifies
i lifi tto:
EX   0
var  X   s 2t
– Mean position over many realizations remains zero
zero.
– Variance of position increases linearly with time.
Random walk in one dimension
• Probabilities of each position in a simple unbiased
random walk, with equal probabilities of moving left
and
d right:
i ht
Position (X)
Time -5s
5s
-4s
4s
-3s
3s -2s
2s -s
s
0
s
2s 3s
4s
5s
t=0
1
t=1
0.5
0.5
t=2
0 25
0.25
05
0.5
0 25
0.25
t=3
0.125
0.375
0.375
0.125
t=4
0.0625
0.25
0.375
0.25
0.0625
t=50
0.031
03
0.156
0
56
0.313
0
3 3
0.313
0
3 3
0.156
0
56
0 03
0.031
• Assumes statistical independence across both
position (x) and time (t)
(t).
Random walk in one dimension
• Random walk on a 1D grid modeled exactly with a
binomial distribution, and approximated by a normal
distribution:
Random walk in one dimension
• Normal approximation:
– Probability that the fish lies in the range  x, x   x 
after
a
te ttime
e t iss given
g e by
by:
2

  x  ts  2 p  1  
1
 x
Pr 
exp 
2
 8ts p 1  p  
2 s 2 p 1  p  t


– Suppose:
• N individuals released at t = 0.
• Carry out independent random walks.
– Then the expected density (concentration) of fish at a
distance x from starting point is:
2

  x  ts  2 p  1  
N

u  x, t  
exp 
2
 8ts p 1  p  
2 s 2 p 1  p  t


Random walk in one dimension
• In the special case of p  12 , density simplifies to:
2


N
x 
u  x, t  
exp  2 
s 2 t
 2ts 
• Based on properties of normal distribution:
– 95% of cases lie within 1.96 standard deviations of
the mean.
– So expect to find, on average, 95% of the fish within
Standard normal distribution
1.96 s t
0.4
0 35
0.35
of the origin at any time t.
50% (z=0.674)
0.3
f(X)
0.25
0.2
0.15
90% (z=1.645)
0.1
95% (z=1.960)
0.05
99% (z=2.576)
0
-3
-2
-1
0
1
z (standard deviation units)
2
3
Random walk in one dimension
• Expected specific densities for unbiased random
walks, based on normal approximation:
p = 0.50; t = 5, 10, 20, 30, 50
0.18
0.16
0.14
Densiity
0.12
01
0.1
0.08
0.06
0.04
0.02
0
-20
-15
-10
-5
0
5
Position (x)
10
15
20
Random walk in one dimension
• Expected specific densities for biased random walks,
based on normal approximation:
p = 0.30; t = 5, 10, 20, 30, 50
p = 0.75; t = 5, 10, 20, 30, 50
0.2
0.25
0.18
0.16
0.2
0.12
Densitty
Densitty
0.14
0.1
0.08
0.15
0.1
0.06
0.04
0.05
0.02
0
-40
-35
-30
-25
-20
-15
-10
-5
0
0
Position (x)
0
5
10
15
20
25
Position (x)
p  Pr( R)
30
35
40
45
Random walk in one dimension
• Can simulate random walks by (pseudo
(pseudo-)randomly
)randomly
generating values {+1, –1} in proportion to the
probabilities p and q
p
q=1-p
p across t steps.
p
– Generate N independent random walks to provide N
predictions of position x at time t.
– Mean, standard deviation and variance of the N
values provide descriptors of the distribution of x.
t = 10
350
t = 10
p = 0.25
N = 1000
Mean = -4.94
Stdev = 2.68
Var = 7.16
300
250
t = 10
p = 0.50
N = 1000
Mean = 0.18
Stdev = 3.14
Var = 9.88
300
p = 0.75
N = 1000
Mean = 4.96
Stdev = 2.78
Var = 7.75
250
200
150
Frequency
200
Frequency
Frequency
250
150
100
150
100
100
50
50
0
-12
200
-10
-8
-6
-4
-2
Position
0
2
4
6
0
-10
50
-8
-6
-4
-2
0
Position
2
4
6
8
10
0
-8
-6
-4
-2
0
2
Position
4
6
8
10
12
Random walk in two dimensions
• Random walk in a plane results in same form of
model and solution:
– But in two variables (x
(x,y)
y) rather than one
one.
– Allows movement in four directions, at mutual right
angles,
g , with specified
p
p
probabilities.
– Examples of unbiased random walks:
t = 100
t = 100
20
30
t = 100
10
15
20
10
5
5
0
y
0
y
y
10
0
-5
-10
-5
-10
-20
-15
-30
-30
-20
-10
0
x
10
20
30
-20
-20
-10
10
-15
-10
-5
0
x
5
10
15
20
-10
-5
0
x
5
10
Random walk in two dimensions
• As with 1D walks, density distributions of final
positions for N walks can be simulated:
px  0.5, p y  0.5
30
px  0.3, p y  0.5
Final positions at t = 50
px  0.7, p y  0.7
Final positions at t = 50
40
Final positions at t = 50
40
30
20
30
20
20
10
10
0
0
y
0
y
y
10
-10
-10
-10
-20
-20
-30
30
-20
20
-30
-40
-30
-30
-20
-10
0
x
10
20
30
-40
-40
-30
-20
-10
0
x
10
20
30
40
-40
-30
-20
-10
0
x
10
20
30
40
Random walk in two dimensions
• Bivariate solution is exactly multinomial, but can be
approximated by bivariate normal distributions.
– Marginal distributions are normal.
– Extends to:
• Correlated random walks.
• Random
R d
walks
lk iin k dimensions
di
i
((multivariate
lti i t normall
distributions).
Bivariate normal distribution
• 5-parameter distribution:
f ( X ,Y ) 
1
2 X  Y 1  
2
e
 X    2  Y    2
 X   X  Y  Y
X
Y
 1 2 
 
 2  

2 (1  )   X
Y 
 X   Y




0.2
0.15
0.1
0.05
0
4
2
0
-2
Y-axis
-4
-4
-3
-2
-1
X-axis
0
1
2
3
4


 
Diffusion in one dimension
• Diffusion: continuous random movement of
molecules or other particles.
• Assume that molecules move along one dimension
(
(x-axis):
i )
– Concentration (density) of molecules at position x at
time t: c(x,
c(x t).
t)
– Number of molecules at time t in the interval (x1, x2):
N  x1 , x2  (t )   c  x, t  dt
d
x2
x1
Diffusion in one dimension
• Molecules move randomly
randomly, so number of molecules
in a given interval changes with time.
• Can express change as difference between net
movement of molecules at the ends of the interval.
• Flux: net number of molecules crossing x from the
left to the right during a time interval of length ∆t:
Flux  J  x, t  t for random movement
• Change in number of molecules in interval  x0 , x0  x 
during the time interval  t , t  t  :
N  x0 , x0 x   t  t   N  x0 , x0 x   t 
 J  x0 , t  t  J  x0  x, t  t
Diffusion in one dimension
• Divide
Di id b
both
th sides
id b
by ∆t and
d llett ∆t → 0:
0
lim
t 0
N x0 , x0 x   t  t   N  x0 , x0 x   t 
t
 J  x0 , t   J  x0  x, t 
• By definition:
N  x0 , x0 x   t  t   N  x0 , x0 x   t  d
lim
 N  x0 , x0 x   t 
t 0
t
dt
d x0 x
 
c  x, t  dx
x
dt 0
• This is the derivative of an integral. When c(x, t) is
sufficiently smooth, can interchange differentiation and
integration.
Diffusion in one dimension
x0 x c  x, t 
d x0 x
• Then:
c  x, t  dx  
dx

x0
dt x0
t
• Putting this all together, we get:
x0 x c  x, t 
dx  J  x0 , t   J  x0  x, t 
x0
t
• Divide both sides by ∆x and take the limit ∆x → 0:
c  x0 , t 
1 x0 x c  x, t 
– Left-hand side: lim
dx 

x
x 0 x
0
t
t
J  x0 , t   J  x0  x, t 
J  x0 , t 
– Right-hand side: lim

x 0
x
x
c  x0 , t 
J  x0 , t 
– Together:

t
x
Diffusion equation
• Combining and simplifying
simplifying, we get the diffusion
equation:
c  x, t 
 2 c  x, t 
c
 2c

D 2
D
2
t
x
t
x
– where D is a p
positive diffusion constant
(in units of distance2/time):
x 

D
2
Conncentration (c)
2t
• In words: rate of change of concentration at a point is
proportional
ti
l tto th
the ‘‘curvature’
t ’ iin th
the relationship
l ti
hi off
concentration to distance.
Position (x)
Diffusion equation
c
 2c
D 2
t
x
• Diffusion
Diff i equation
ti is
i ubiquitous
bi it
iin bi
biology:
l
– Brownian motion and osmosis: descriptions of random
movements of particles.
– Change in allele frequencies due to random genetic drift.
– Invasion of alien species into new habitats.
– Chemotaxis: movement of organisms along chemical
gradients.
– Pattern formation in developing embryos along morphogen
gradients.
– Physics: heat equation describing diffusion of heat
through a solid bar (c = temperature).
– Represents a null hypothesis: must be rejected to show
that movement or change is non-random.
Diffusion equation
c  x, t 
 2 c  x, t 
D
t
x 2
c  x, t 
• For diffusion, we get Fick’s law: J  x, t    D
x
– Flux is linearly proportional to the change in
concentration along the position axis
axis.
– Negative sign means that the net movement of
molecules is from regions of high concentration to region
of low concentration.
– If no change in concentration, net movement is zero.
Diffusion constant
• Diffusion constant,
constant =diffusion coefficient,
coefficient
=diffusivity: rate at which a diffusing substance is
transported
p
between opposite
pp
faces of a unit cube
of a system when there is unit concentration
difference between them.
Diffusion in one dimension
• F
For mostt purposes, the
th diffusion
diff i equation
ti can be
b
solved analytically:
c  x, t 
 2 c  x, t 
D
t
x 2
 x2 
1
c  x, t  
exp  

4 Dt
 4 Dt 
• Compare with equation for the normal distribution:
2

x   

1
f  x 
exp  

2
2
 2


  0,   2Dt
Diffusion in one dimension
• Rates of change of c(x,t) with time and curvature of c.
c(x, t)
t = 1, 2, 3, 4, 10, 20, 30, 40
x
Diffusion in two dimensions
• Diffusion equation generalizes easily to higher
dimensions.
• For diffusion in a plane:
  2 c  x , y , t   2 c  x, y , t  
c  x, y, t 
 D


2
2
t
x
y


2
2



x
y


1

exp  
c  x, y , t  
4 Dt 
4 Dt


• General prediction: area is linearly proportional to time.
• Diffusion constant D is independent of number of
dimensions.
– Measures ability of particles to move through medium
medium.
Diffusion, random walk,
and normal distributions
• Normal distribution arises in two different ways:
– 1D random walk:
• Described exactly by a binomial distribution of stepwise
movements.
• Approximated by a normal distribution when N is large.
• Extrapolates to multiple dimensions.
– 1D diffusion equation:
• Based on concept of flux: net number of molecules
crossing x from the left to the right during a time interval
of length ∆t.
• Solution of the diffusion equation yields a model
equivalent to a truncated normal distribution.
• Extrapolates to multiple dimensions.
Example: spread of muskrats
• One of first applications of diffusion equation in
ecology (Skellam 1951):
– Several muskrats accidentally released in 1905 in
central Germany.
– Spread
p
to cover most of Germany
y and Austria by
y 1927.
– Rate of spread is consistent with diffusion model:
area
Example: larval wanderings
• M
Movementt off flatworm
fl t
larvae
l
(Broadbent
(B db t and
d
Kendall, 1953):
– Larvae placed at center of series of concentric circles.
– At each time interval, counted numbers of larvae
within circles:
–E
Expressed
d 2D diff
diffusion
i model
d l
in terms of polar coordinates
such that linearity of terms is
consistent
i
with
i h model:
d l
Parenthetical topic: polar coordinates
• Cartesian and polar coordinate systems:
r  x2  y 2
y
  arctan
x
x  r  cos 
y  r  sin 
• Typically used in models of animal dispersion.