Introduction
In general, the passive movement of particles, or diffusion, can be explained in one
simple equation, Fick’s First Law of Diffusion:
F = -D ∙ A ∙ dC/dX
F = Flux
D = Diffusivity Coefficient
A = Surface area that the material is diffusing
through
dC/dX = Concentration Gradient
The concentration gradient describes how quickly the concentration changes, as the
particles move a specified distance. The negative sign is incorporated into the equation because
particles will tend to move in a direction opposite of the concentration gradient, thus decreasing
the gradient as they move. Fick’s Law can be understood intuitively—if the diffusivity of the
particles is increased (or the particles are allowed to moved more freely), then the rate of flow
should increase. Also, if the area available for diffusion increases, the flux should increase.
Lastly, if the concentration gradient increases, the flux should increase. Fick’s First Law of
Diffusion summarizes these logical relationships between diffusivity, area, concentration
gradient, and flux.
Diffusion can be found in nearly all aspects of life—from the movement of heat
throughout a room to the movement of nutrients through the bloodstream. It controls the
environment around us, as well as the environment within our bodies. For example, across the
membranes of excitable cells there exists a large amount of potassium outside of the cell and
sodium inside the cell, establishing high concentration gradients. When certain channels allow
the ions to pass through the cell’s membrane, potassium enters the cell and sodium leaves the
cell, moving down their respective concentration gradients. This movement is governed by
Fick’s Law; the ions move “down” their gradients in the negative direction, simultaneously
decreasing the magnitude of the gradient and decreasing the flux. This process of facilitated
diffusion causes excitable cells, such as neurons, to pass signals through the body via action
potentials.
2
In addition to allowing signal transmission, diffusion is used in the transportation of
oxygen from the lungs to the alveoli—a process also explained by Fick’s Law. In the lungs,
oxygen passively diffuses across a membrane into the bloodstream. The branched form of the
lungs allows for an exponential increase in surface area, thus allowing for an increase in flux and
more effective movement of oxygen into the bloodstream. Relating this phenomenon to Fick’s
Law, as A increases, F also increases.
As shown in the aforementioned situations, diffusion is an integral part of living systems.
However, some questions arise. Particles have no way of “knowing” which way they should
diffuse in order to follow Fick’s Law. In fact, diffusion is random by definition. The purpose of
this lab is to illustrate and describe the movement of particles through computer simulations and
explain why random diffusion moves from high to low concentrations.
Materials and Methods
The four simulations were conducted using MATLAB software version 6.5; graphs were
generated in MATLAB or recreated using Microsoft Excel. The instructions and commands were
downloaded from the WebCT site for BIO 201,
http://one.drexel.edu/cp/school/webctFrame?course_id=13533200415
Results
In the following simulations, several assumptions were made to facilitate the demonstration of
diffusion. In each instance after each collision, the molecules moved the same distance, the fixed
free path length, and in random directions and two dimensions. All collisions were perfectly
elastic so that kinetic energy was conserved. In simulations involving many particles, it is
assumed that all of these thousands of molecules occupied the same point (0,0) at the start, and
moved randomly in only two dimensions. Time is represented as the number of collision times
with either molecules in the surrounding medium or other particles participating in the
simulation. Since the distance traveled between each collision is assumed to be the same, and
3
since kinetic energy conserved and thus velocity can be assumed to be the same, the number of
collisions is equal to time.
Simulation 1: Path of a Single Molecule- The Random Walk
Graph 1: Path of a Molecule beginning at the Point (0,0)
Graph 1 shows the path that a single molecule took while moving randomly through a medium,
such as water.
Graph 2: Distance Traveled over Time
4
Graph 2 shows the distance that the single molecule traveled from the start of the simulation at
the point (0,0), as a function of time.
Simulation 2: Movements of Numerous Molecules
Graph 3: Location of Each Molecule at the End of the Simulation
Graph 3 shows the positions of each molecule at the end of the simulation, after they traveled
randomly from the start point of (0,0).
Graph 4: Average Distance Traveled by Each Molecule over Time
5
The number average of the distance traveled by each of the thousands of molecules in this
simulation is plotted versus time to show the general relationship between distance from the start
point and time.
Graph 5: Distance traveled by Different Subsets of Molecules over Time
Graph 5 shows the maximum distance traveled by a molecule in various subsets of the group of
molecules. Though every molecule begins at the origin and travels the same distance between
each collision, they travel different amounts of distance, which is related to their proximity to the
origin at a given time-point. The pink line at the bottom of the graph represents the maximum
distance traveled by a molecule in the subset encompassing 5% of all molecules, specifically the
ones closest to the origin. The red line above that represents those 5% plus another 5%, so that
the red line represents the 10% of the entire group that are closest to the origin. Thus, the entire
group of molecules can be thought of as concentric circles encompassing all the molecules closer
than it to the origin. Therefore, the pink line at the top of the graph represents 95% of all the
molecules.
6
Simulation 3: Diffusion of Molecules in a Box
Graph 6: Sequence of Plots of Positions of Molecules over Time
This figure shows the movement of the molecules across the box over time. Each plot represents
a different time point, with the upper left-hand box representing time=0, the box to its right is the
second time point, etc, so that the box at the bottom right-hand corner is the 9th time point. In
each plot, the x-y position of each molecule is plotted as if the molecules were frozen in time, so
that the net movement of particles can be seen over the nine plots.
This simulation produced two other graphs; one illustrating the movement of many molecules
and the other generalizing the fraction of molecules in the left and right side of the box at a given
point in time. However, the graphs produced in MATLAB were unclear and two new graphs
were created from the provided Microsoft Excel files. The two graphs are shown below.
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Graph 7: Histogram Curve of the Movement of Molecules
Relative Frequency of Molecules
1
5
Movement of Molecules From Left to Right
10
20
0.120
50
100
200
0.100
500
1000
2000
0.080
3000
4000
5000
0.060
6000
7000
8000
0.040
9000
10000
20000
0.020
30000
40000
0.000
50000
60000
70000
80000
X Position
90000
100000
Graph 7 represents the movement of particles as influenced by the number of collisions each
particle experienced during the simulation.
85
65
45
25
5
5
-1
5
-3
5
-5
5
-7
5
-9
Graph 8: Exponential Curve of the Movement of Molecules
Movement of Molecules From Left to Right
1.000
0.800
0.600
0.400
0.200
1E+05
80000
60000
40000
20000
9000
7000
5000
3000
1000
200
50
10
0.000
1
Relative Frequency of Molecules
1.200
Time (# of collisions)
Fraction of molecules (Left box)
Fraction of molecules (Right box)
8
Graph 8 shows the fraction of molecules in each side of the box. At the start of the simulation,
100% of the molecules were in the left box, and as the simulation progressed, the fraction of
molecules in each side converged at 50%--an even distribution.
Simulation 4: Diffusion of Molecules Between 2 Boxes Connected by a “Pipe”
Graph 9: Sequence of Plots of Positions of Molecules over Time
The circles in the above Graph 9 represent particles moving between two chambers by way of a
pipe. Each of the 6 small graphs here represents a different moment in time—at 50, 2000, 7000,
50000, 80000, and 400000 collisions.
In Simulation #4, the graphs produced by MATLAB were unclear. Thus, a new set of data
points was provided to create clear graphs. Graphs were made for data points from 4 separate
systems with varying pipe diameters of 4, 8, 12, and 16. For each pipe size, two graphs were
made, similar to the graphs produced in Simulation #3—a histogram graph and an exponential
curve.
9
Graph 10: Movement of Molecules Between Two Chambers Through a Pipe of Diameter 4
Movement of Molecules Through Pipe
Length of Pipe=20
Diameter of Pipe=4
1
5
10
20
0.09
50
100
0.08
200
500
1000
Relative Frequency of Molecules
0.07
2000
3000
4000
0.06
5000
6000
0.05
7000
8000
9000
0.04
10000
20000
0.03
30000
40000
50000
0.02
60000
70000
80000
0.01
90000
70
62
54
46
38
30
22
14
6
-2
-1
0
-1
8
-2
6
-3
4
300000
-4
2
200000
-5
0
100000
0
X Position
400000
500000
Graph 10 shows the distribution of molecules at a given time. The hump on the left side of the
graph represents the left chamber, the right hump represents the right chamber and the valley
between the two represents the pipe between the two chambers. On the left side of the graph, the
span of point representing the molecules on the left hand side illustrates the concentration
gradient. At first, there is a high relative frequency of molecules, and as the system moves town
equilibrium the number of molecules on the left side decreases. This histogram aptly
demonstrates the movement of the molecules down their concentration gradient. This pattern is
true for the three remaining histograms in this simulation.
10
Graph 11: Exponential Curve of Molecular Movement Between Chambers (Pipe Diameter=4)
Movement of Molecules Through Pipe
1.20
1.00
Relative frequency of
molecules (Left box)
0.80
Relative frequency of
molecules (Pipe)
0.60
Relative frequency of
molecules (Right box)
0.40
0.20
0
00
0
0
50
00
0
00
0
20
80
00
0
50
00
0
20
00
80
00
50
0
00
20
20
20
0.00
1
Relative Frequency of Molecules
Length of Pipe=20
Diameter of Pipe=4
Time (# of collisions)
Graph 11 illustrates the movement from the left chamber to the right chamber over time. It
shows the relative distribution of the molecules as the system reaches equilibrium.
11
Graph 12: Movement of Molecules Between Two Chambers Through a Pipe of Diameter 8
1
Movement of Molecules Through Pipe
5
Length of Pipe=20
Diameter of Pipe=8
10
20
50
0.1
100
200
0.09
500
1000
Relative Frequency of Molecules
0.08
2000
3000
4000
0.07
5000
6000
0.06
7000
8000
0.05
9000
10000
0.04
20000
30000
0.03
40000
50000
60000
0.02
70000
80000
0.01
90000
100000
X Position
70
62
54
46
38
30
22
14
6
200000
-2
-5
0
-4
2
-3
4
-2
6
-1
8
-1
0
0
300000
400000
500000
Graph 12 is another histogram similar to the first, except that the pipe in this system is double the
diameter of the first. Compared to the previous histogram, it can be seen that the number of
molecules residing in the pipe at equilibrium has increased, due to the increase in diameter of the
pipe.
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Graph 13: Exponential Curve of Molecular Movement Between Chambers (Pipe Diameter=8)
Movement of Molecules Through Pipe
1.20
1.00
0.80
Relative frequency of
molecules (Left box)
0.60
Relative frequency of
molecules (Pipe)
0.40
Relative frequency of
molecules (Right box)
0.20
500000
200000
80000
50000
20000
8000
5000
2000
200
20
0.00
1
Relative Frequency of Molecules
Length of Pipe=20
Diameter of Pipe=8
Time (x of collisions)
Graph 13 shows the movement of molecules through the pipe of diameter 8. Similar to Graph 8
of Simulation #3, the relative frequency of molecules in the left and right chamber is
approximately 0.50 as the system reaches equilibrium. However, because there is a pipe
connecting the two sides in this system, some of the molecules will still reside in the pipe, shown
by the fuchsia line in the graph above.
13
Graph 14: Movement of Molecules Between Two Chambers Through a Pipe of Diameter 12
Movement of Molecules Through Pipe
Length of Pipe=20
Diameter of Pipe=12
1
0.1
5
10
20
0.09
50
100
0.08
200
Relative Frequency of Molecules
500
1000
0.07
2000
3000
4000
0.06
5000
6000
0.05
7000
8000
9000
0.04
10000
20000
30000
0.03
40000
50000
0.02
60000
70000
80000
0.01
90000
100000
200000
0
X Position
70
62
54
46
38
30
22
14
6
-2
-1
8
-1
0
-5
0
-4
2
-3
4
-2
6
300000
400000
500000
In Graph 14, the molecules are now moving between the two chambers through a pipe with a
diameter of 12.
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Graph 15: Exponential Curve of Molecular Movement Between Chambers (Pipe Diameter=12)
Movement of Molecules Through Pipe
Length of Pipe=20
Diameter of Pipe=12
Relative Frequency of Molecules
1.20
1.00
0.80
Relative frequency of
molecules (Left box)
0.60
Relative frequency of
molecules (Pipe)
0.40
Relative frequency of
molecules (Right box)
0.20
500000
200000
80000
50000
20000
8000
5000
2000
200
20
1
0.00
Time (# of collisions)
Graph 15 illustrates this system moving towards equilibrium. Following the trend observed in
Graph 13, the relative frequency of molecules within the pipe continues to increase with increase
in diameter.
15
Graph 16: Movement of Molecules Between Two Chambers Through a Pipe of Diameter 16
Movement of Molecules Through Pipe
Length of Pipe=20
Diameter of Pipe=16
0.1
0.09
Relative Frequency of Molecules
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
X Position
70
62
54
46
38
30
22
14
6
-2
-1
0
-1
8
-2
6
-3
4
-4
2
-5
0
0
1
5
10
20
50
100
200
500
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
20000
30000
40000
50000
60000
70000
80000
90000
100000
200000
300000
400000
500000
As seen in the previous histograms and exponential curves, the number of molecules in the pipe
has increased compared to the previous graphs. It is also important to note that the system is
moving closer toward equilibrium at a lesser number of collisions. For example, at 20000
collisions (represented by the pea green line in the graph above) the system is already on the
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verge of reaching equilibrium. In previous graphs, the system was approximately half way to
equilibrium at 20000 collisions.
Graph 17: Exponential Curve of Molecular Movement Between Chambers (Pipe Diameter=16)
Movement of Particles Through Pipe
1.20
1.00
0.80
Relative frequency of
molecules (Left box)
0.60
Relative frequency of
molecules (Pipe)
0.40
Relative frequency of
molecules (Right box)
0.20
0
0
00
0
50
00
0
20
00
0
80
00
0
50
00
0
00
20
80
00
50
0
00
20
20
20
0.00
1
Relative Frequency of Particles
Length of Pipe=20
Diameter of Pipe=16
Time (# of collisions)
Graph 17 is a clear illustration of changes in the system. At 2000 collisions there already exists a
large number of molecules in the pipe. The system reaches equilibrium by approximately 30000
collisions, which is considerably earlier than the time it took previous systems to reach
equilibrium.
Discussion
Simulation 1: Path of a Single Molecule- The Random Walk
The random walk taken by this single molecule is the same path taken for each molecule
throughout these simulations. Graph 1 indicates that there is no clear directionality in the
molecule’s movement. The dense areas of movement in Graph 1 also show that all movement is
random; if it were not, the lines would be more distinct. Furthermore, a clear direction would be
17
preferentially chosen if the movement had directionality. This idea is supported upon
repetitions of the experiment, with no clear emerging patterns. In each run of the simulation, the
molecule always ends up in a different location, across the four quadrants of the coordinate
system. Thus, the movement of the particle is completely random and not directional.
The only trend that can be seen throughout repeated simulations is that the particle
moves—in any direction—further from the origin. This idea is graphically represented in Graph
2, in which the distance that the particle traveled increased with increasing collisions between the
molecule and the surrounding medium. There is a positive correlation between the number of
collisions and the distance from the origin, and it is seen throughout repeated simulations, despite
variances in the direction that the particle takes.
In this simulation, several important assumptions are made. For simplification purposes, a
fixed length is assigned to the distance traveled between each collision and the velocity does not
change, as if each collision was perfectly elastic and momentum completely conserved.
Furthermore, the surrounding medium is assumed to be perfectly homogenous. In biological
systems, this is never the case. For a single molecule traveling through a medium, the change in
momentum following each collision with another molecule would probably be enough to stop or
at least significantly impede the movement of the molecule. The distance traveled from the
origin would not be so great, and the path would be altered depending on collisions with
differently sized molecules, and different open pathways for the molecule to take.
Simulation 2: Movements of Numerous Molecules
Graph 3 shows the movement of many particles which start at the same point. It is
obvious that their paths have no directionality because they end at many different points all over
the graph, and one area is not preferred to another.
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The mean distance from the origin increases with time, shown in Graph 4, and because
distance is directly proportional to time on a logarithmic scale, the mathematical theory that
distance is proportional to the square of time is supported.
Graph 5 shows the correlation between maximum distance traveled and proximity to the
origin. The pink line at the bottom of the graph represents the 5% of the molecules closest to the
origin. These molecules appear to have traveled a negative distance initially; this is due to the
high amount of molecules near the origin pushing these molecules closer to the origin, rather
than allowing them to move away in the positive y direction. This phenomenon appears to be
true for the 10% group as well, but all the other groups behave predictably, traveling further and
further from the origin as time goes on. Graph 5 shows the maximum distance traveled by a
molecule in each subset, explaining why the group of molecules furthest from the origin travels
the furthest; these molecules have more room to travel than molecules closer to the origin.
The wave front of the diffusing mass moves randomly in all directions, and in those
directions opposite from collisions with other molecules. Molecules on the boundary of the mass
collide with those beneath them (i.e. closer to the origin) and move away from the collisions and
thus away from the mass. In this manner, the molecules on the outskirts of the mass move away
from the origin first, and allow for movement of molecules behind them.
Simulation 3: Diffusion of Molecules in a Box
Simulation 3 show the movement of molecules from the left side of the box, as the
molecules spread through the entire box to reach equilibrium, as shown by Graph 8. At
equilibrium, the distribution is 50/50 with half of the molecules on the left side of the box and
the other half on the right side. Statistically, this distribution is expected because there are the
most possible distributions of molecules that result in a 50/50 spread. For example, it is not
likely that all the molecules congregate in one corner because there is only one distribution that
19
would result in that situation. It becomes more likely for all of the molecules to be spread all
the way out because there exist more locations for the molecules to reside in if the molecules
diffuse throughout the entire box.
In Graph 8, it is shown how quickly the relative frequency of molecules for each side
approaches 0.50. The system seems to be in equilibrium around 80000 collisions. Qualitatively,
it can be determined how quickly the system reaches equilibrium by taking the slope, or
derivative, of the lines shown in Graph 8. The system appears to be nearly equilibrium at an
exponential rate, more specifically from 10000 collisions to the end of the experiment. Before
10000 collisions the rate of diffusion seems to be very roughly exponential. Again, the rate of
diffusion can be determined at any point by taking the derivative of the function at any given
point in time. This method of finding the slope provides a clear and quantitative way of
comparing multiple systems.
The rate at which a system reaches equilibrium is dependent on the mean free path and
velocity. As the mean free path decreases, the rate at which the system will reach equilibrium
increases. This inverse relationship is true because if the mean free path decreases, then the
molecule is experiencing more collisions, which cause the particles to disperse. As the velocity
increases, the rate at which a system will reach equilibrium increases because with more
movement comes more collisions as well. If the dimensions of the box were smaller,
equilibrium would be approached more rapidly because smaller dimensions increase the
probability that a molecule collide with a wall. In addition, smaller dimensions would give the
molecules less distance to travel, causing them to reach a 50/50 spread more rapidly. More
collisions cause the molecules to be redirected more frequently, spreading them across the given
area. These hypotheses could be testing by changing the parameters in the program and
rerunning them to see how quickly each system approaches equilibrium.
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The histogram in Graph 7 illustrates the positions of the molecules over definitive
periods in time. The intermediate times show the general trend as the molecules move from the
left side to the right side. In the beginning of the simulation, the molecules diffuse more rapidly
than at the end of the simulation. For example, between 40000 collisions and 100000 collisions,
there is barely any change in the distribution of molecules. At this point, the molecules have
already advanced down their concentration gradient. The slopes of the lines at the beginning of
the simulation are considerably steeper than the lines at the end of the simulation, suggesting a
decrease in the rate of diffusion as the simulation progresses. This pattern bolsters the idea of
diffusion occurring down a concentration gradient and illustrates the importance of collisions
between the molecules. Toward the end of the simulation, the molecules become increasingly
spread out, causing fewer collisions between molecules and a slower rate of diffusion.
Simulation 4: Diffusion of Molecules Between 2 Boxes Connected by a “Pipe”
At equilibrium, the distributions for all four systems with varying pipe diameters are
generally similar—slightly less than 0.50 in both chambers and a small percent of the molecules
residing in the pipe. The trends in this simulation are similar to that in Simulation 3, but in this
simulation there are a small percentage of molecules that remain in the pipe when the system has
reached equilibrium. The number of molecules in the pipe increases with increasing diameter
because a larger diameter increases the probability of a molecule entering the pipe. Furthermore,
a larger diameter causes an increase in the volume of the pipe, allowing a larger pipe to hold a
larger percentage of the molecules.
The rate at which the system approaches equilibrium can be determined by taking the
derivative of the lines in the exponential curve. This rate changes over time due to the change in
concentration gradient. Therefore, when the derivative is calculated, any point in time can be
plugged into the equation for the derivative to determine the rate of diffusion at that point in
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time. In addition to this method, the rate of diffusion can also be calculated using Fick’s Law,
as described in the introduction.
The diameter of the pipe heavily influences how quickly the system will reach
equilibrium. A larger diameter allows more molecules to pass from one chamber to another,
while a smaller diameter impedes this motion. This phenomenon can be seen in Graphs 11, 13,
15, and 17. In Graph 11, the diameter of the pipe is 4 and it takes approximately 500000
collisions for the system to reach equilibrium. Conversely, Graph 17 shows the system with two
chambers connected by a pipe with a diameter of 16. For this second system, it took
approximately 70000 collisions to reach equilibrium. In order to compare pipe diameter and rate
of diffusion, one could plot pipe diameter on the x-axis and time it took to reach equilibrium on
the y-axis. The resulting plot would illustrate an inverse relationship. If one wanted to compare
pipe length and rate of diffusion, a similar plot could be made. However, this second plot would
be a direct correlation; as pipe length increases, time it takes the system to reach equilibrium
would also increase.
Free path length and velocity of the molecules also contribute to the rate of diffusion. If
the molecules have a very short path length, they will experience multiple collisions. These
collisions are the driving force in diffusion. Similarly, if the molecules have a higher velocity,
there will be more collisions and a higher rate of diffusion. Thus, decreasing free path length
and increasing velocity increase the rate at which the systems will reach equilibrium.
In this simulation, the rate of approach to equilibrium is not completely a fair assessment
of the rate of diffusive transfer. In some ways, the assessment can be seen as far because
molecules are merely diffusing through a region of space, including the pipe. However, the
dimensions of the pipe in this situation have an extremely large effect on the rate of approach to
equilibrium. Because the pipe impedes diffusion to such a large degree, it must be taken into
consideration when assessing how quickly the molecules diffuse.