Diffusion and its Effect on the Scale of Life

By:

Kevin Clark &

Vicky Romberger

​

Mentor: Greg Herschlag

Biological Questions

• Why were insects several feet long millions of years ago,

but are only a few inches long now?

• Why are cells on the scale of microns, without

significant variation between different types of cells?

​

Biological Questions

• Diffusion
• Probabilistic model behind process
• Brownian Motion
• Rates of diffusive processes
• Size limitations of bacterial cells
• Limitations on size of insects
• Potential explanation to the size difference between now and millions of years ago?

Brownian Motion

• Robert Brown - 1827
• Witnessed “random” movement of pollen grains in water

Random Walks

• One dimensional random walk
• Particle has equal probability of moving left or right by one unit
• In two or three dimensions, general idea is the same

Probability behind Random Walks

• After one step:
• Expected Value of particle position is:

​

​

​

​

​

​

​

• Variance of particle position is:

​

​

​

​

​

​

​

​

​

​

​

Probability behind Random Walks

• After two steps:
• Expected value of particle position is:

​

​

​

​

​

• Variance of particle position is:

​

Probability behind Random Walks

• After three steps:
• Expected value of particle position is:

​

​

​

​

​

• Variance of particle position is:

​

​

Probability behind Random Walks

• This pattern continues for infinitely large number of steps
• Expected value of particle position is always zero
• Variance of particle position increases linearly and equals the average squared step size multiplied by the number of steps taken
• , where M is the number of steps taken, and L is the average step size

Back to Brownian Motion

• In 1 cubic cm of water, there are about 10^22 water molecules, all moving at about 1000 m/s
• This gives an average spacing of about 1 nm between water molecules
• Therefore, each water molecule experiences 10^12 collisions per second, on average

Back to Brownian Motion

• The pollen grains that Brown looked at are thousands of times bigger than a water molecule and experiences millions of collisions per second
• How are we able to see the pollen grain moving, since the collisions of the water molecules with the pollen grain should cancel each other out?

Back to Brownian Motion

• The answer to this question was published in 1905 by Albert Einstein
• On average, the number of water molecules hitting the pollen grain from the left and right will cancel each other out
• However, every several million collisions, hundreds more water molecules will hit the pollen grain from the left than from the right

Diffusion

• In diffusion, molecules undergo a “random walk”
• As we know,
• For time step , , where t is the total time elapsed
• So we have
• Defining , we get:

​

• This is known as the one-dimensional Diffusion Law, where D is known as the diffusion constant of the process

Fick’s Law

• Flux - the number of particles passing through a space per unit area per unit time. Denoted by j
• Assume a steady state
• Consider a collection of particles in a series of “boxes”
• Then the total flux at the boundary between the first two boxes equals the number of particles going from box 1 into box 2 (N(x)/2) minus the number of particles going from box 2 to box 1 (N(x + x )/2) per unit area ( ) per unit time ( )

Fick’s Law

Diffusion Equation / Heat Equation

Diffusion and Bacterial Metabolism

• Consider a spherical bacteria of radius R suspended in a large lake
• Assume that the concentration of oxygen in the lake, far from the bacteria, is C( ) = C0
• Assume that the bacteria uses the oxygen as soon as it reaches the surface of the bacteria, so C(R) = 0
• Finally, assume that we are at a steady-state equilibrium, so the concentration of oxygen at any given point in not changing. In other words, dc/dt = 0

Diffusion and Bacterial Metabolism

• Consider a series of concentric spheres surrounding the bacteria. Since we are at steady state conditions, oxygen is not building up anywhere, so the amount of particles entering any given shell must equal the amount of particles leaving the shell.
• In other words, the flux multiplied by the surface area of the shell must be constant, independent of the radius r of the shell. Call this constant K
• So

Diffusion and Bacterial Metabolism

• By Fick’s Law, we have:

Diffusion and Bacterial Metabolism

Limitation on Bacterial Size

• In 300K water, Doxygen = 0.00004 cm^2/s
• Metabolic rate of bacteria = 0.02 mol O2/kg*s
• Concentration of oxygen in water = 2*10^(-5) g/cm^3

Limitation of Bacterial Size

Limitation of Bacterial Size

Limitations of Bacterial Size

Could an Insect have been 10 times larger in prehistoric times than they are today??