Cell Process
1. Cell and transport phenomena
Liang Yu
Department of Biological Systems Engineering
Washington State University
02. 12. 2013
Main topics
• Cell and transport phenomena
• High performance computation
• Metabolic reactions and C-13 validation
• Enzyme and molecular simulation
Introduction
• Basic knowledge in cell
– the basic structural and functional unit of all known living
organisms
– There are two types of cells: eukaryotic and prokaryotic
– Organisms can be classified as unicellular (consisting of a
single cell; including most bacteria) or multicellular
(including plants and animals)
– Most eukaryotic cells are between 10 and 100 µm while
prokaryotic cells are between 1 and 5 µm
Prokaryotic cell
 Smaller than a eukaryotic cell
 Lacking a nucleus and most of the other
organelles of eukaryotes
 Two groups: bacteria and archaea
 Covered with cell wall and plasma membrane
 Membrane is a selective barrier
 Largest molecules cross the membrane are
DNA fragments and low-molecular-weight
proteins
 Flagella and pili are structures made of
proteins that facilitate movement and
communication between cells
Eukaryotic cell
• Plants, animals, fungi, slime moulds, protozoa, and algae are
all eukaryotic
• Plant and fungal cells have a cell wall
• Membrane separates and protects a cell from its surrounding
environment
• Major difference between prokaryotes and eukaryotes is that
eukaryotic cells contain membrane-bound compartments in
which specific metabolic activities take place
Plant and animal cell
Cell membrane
Transport mechanisms
1. Passive diffusion and osmosis: Some substances (small molecules, ions) such
as carbon dioxide (CO2), oxygen (O2), and water
2. Transmembrane protein channels and transporters: Nutrients, such as sugars or
amino acids, and certain products of metabolism
3. Endocytosis: internalizing solid particles (cell eating or phagocytosis), small
molecules and ions (cell drinking or pinocytosis), and macromolecules.
Endocytosis requires energy and is thus a form of active transport.
4. Exocytosis: remove undigested residues of substances, secrete substances such
as hormones and enzymes, and transport a substance completely across a
cellular barrier
Examples for cell simulation
Protein-protein interactions
Trajectories of discrete signaling proteins
Molecular dynamics - Monte Carlo simulation
http://www.vis.uni-stuttgart.de/~falkmn/research.html
Examples for cell simulation
Electron micrograph of
the oxytactic bacterium
B. subtilis
Flow fields around bacteria
http://www.maths.gla.ac.uk/~mab/phd.html
N.A. Hilla, T.J. Pedleyb. Bioconvection. Fluid Dynamics
Research. 2005,37(1–2): 1–20
Major constitutive equations
A constitutive equation is a relation between two physical quantities
(especially kinetic quantities as related to kinematic quantities) that is
specific to a material or substance, and approximates the response of that
material to external stimuli, usually as applied fields or forces.
Biological
System
Mass transfer
Momentum transfer
Energy transfer
Chemical reaction
Bio-information transfer
Transport phenomena
Chemical
Engineering
Theoretical fundamentals for models
Conservation of mass
Conservation of momentum
Conservation of energy (First law of thermodynamics)
dU   Q   W
Principle of minimum energy (Second law of thermodynamics)
 S 

 0
 X U
and
 2S 
 X 2   0

U
at equilibrium
Definition of transport phenomena
• Diffusion
– Random motion of molecules that arises
from thermal energy transferred by
molecular collisions
• Convection
– Mechanism of transport resulting from the
bulk motion of fluids
Diffusion
 Collision between molecules occur trillions of times per second
 A molecular diffusion speed depends on its size and shape, the
temperature, and the fluid viscosity, a property that reflects the
resistance to flow
 In spite of the random nature of these collisions, net motion of
molecules results
 A macroscopic consequence of random molecular motion is the
phenomenon that diffusing molecules move from regions of higher
concentration to regions of lower concentration.
Fick’s first law of diffusion for dilute solutions
• Diffusion flux is proportional to the gradient of the
concentration
J i,x
Ci
  Dij
x
J is the "diffusion flux" [(amount of substance) per unit area per unit
time], example (mol/ m2 s)
C is the concentration in dimensions of [(amount of substance)
length−3], example (mol/m3)
D is the diffusion coefficient in dimensions of [length2 time−1],
example (m2/s)
x is the position [length], example (m)
Diffusion in concentrated solution
J i,x

Ci  d  i
  Dij
xi
xi   i xi 

 i  dxi

 d (ln  i ) 
  DijC  1 
 xi
 d (ln xi ) 
For constant density, Cxi  Ci
Dapp
 d (ln  i ) 
 Dij  1 

d
(ln
x
)
i 

Dapp is the apparent diffusion coefficient
γ is the activity coefficient that acounts for solute-solute interaction
Diffusion coefficient, D
(Stockes-Einstein equation)
• This macroscopic, average constant is called a
diffusion coefficient
• It is a function of temperature and pressure
k BT
Driving "Potential" Chemical Potential Gradient
D


6 r0
Resistance
Stokes Drag Terms
Used for sphere
T = temperature in Kelvin
kB = Boltzmann’s constant = 1.38054 x 10-23 J K-1
r0 = molecular radius of the solute
Diffusion Coefficient
Range of Values for the Binary Diffusion Coefficient,
Dij, at Room Temperature
Diffusing Quantity
Diffusion coefficient, cm2s-1
Gases in gases
0.1 to 0.5
Gases in liquids
1 ×10-7 to 7 ×10-5
Small molecules in liquids
1 ×10-5
Protein in liquids
1 ×10-7 to 7 ×10-7
Protein in tissues
1 ×10-10 to 1 ×10-7
Lipids in lipid membranes
1 ×10-9
Protein in lipid membranes
1 ×10-12 to 7 ×10-10
Diffusion in 1, 2 & 3D
Ways that a bacterium might get
small, dissolved molecules
• Without the aid of hydrolytic enzymes; wait for them
to arrive; go hunting or grazing
• Wear hydrolytic enzymes on its “sleeves”
(immobilized enzymes)
• Send hydrolytic enzymes out on their own (freely
released exoenzymes); collect the products
• Send out a recognizable chelator to catch them and
make them both recognizable and transportable
across the cell membrane (called a siderophore)
E. Coli swimming
Could a bacterium get more small
nutrients by swimming?
Typical swimming speed, u = 50 μm s-1
Swimming encounter flux is πr02Cinf u
Ratio of swimming encounter/diffusion flux < 6.25 x 10-3
An organism of this size cannot outrun diffusion.
http://brodylab.eng.uci.edu/~jpbrody/reynolds/lowpurcell.html
Convection
• Move molecules or cell by fluid motion
• Gases and liquids are fluids that flow following
the application of forces, such as gravity,
pressure, or shearing forces
• The resulting application of a force upon a
surface is characterized in terms of a stress
• Convection is affected by fluid viscosity and
density which are thermodynamic function of
temperature and pressure
Flow pattern and Reynolds number
•
•
•
•
•
For a specific solid geometry
In a specific flow geometry
Whenever Re is similar
The flow pattern is similar
That’s why you can scale model a
fomenter and not get too surprised
• Given geometric similitude, holding Re
constant assures dynamic similitude
Reynolds number in cell-scale
inertial forces  u 2 L  uL
Re 


2
viscous forces u L

For objects moving at the same speed, the relative
significance of viscous and initial forces depends
upon size. For example, for a fish 5 cm long moving in
water at relatively slow speed of 1 cm/s, the Re is
5000. If a white blood cell with a diameter of 10 µm
moves at the seep of our hypothetical fish, then the
Re is 0.1 and viscous force dominate.
Re also represents the ratio of momentum transport
by convection to momentum transport by viscous
diffusion.
Kinematic viscosity
Range of Values for Viscosity, Density, and Kinematic viscosity at Room
Temperature
Viscosity, µ, g cm-1s-1
Density, ρ, g cm-3
Kinematic viscosity
ν=µ/ρ, cm2s-1
Gases
10-4
0.001
0.1
Liquids water
0.01
1.0
0.01
10
1
10
0.03
1.2
0.025
Glycerol
Blood
 Movement of momentum through the fluid can be regarded as
analogous to diffusion
 Kinematic viscosity is a measure of the efficiency of momentum
transport
Relations for momentum, mass and energy
Relations between flux and gradients for molecular transport
Molecular
transport
mechanism
Momentum
Mass
Energy
Flux
Gradient
Coefficient of
proportionality
Shear stress
Velocity
Viscosity
Mass or molar flux
Concentration
Diffusion coefficient
Heat flux
Temperature
Thermal conductivity
Transport equations
 z,x
 u x
 
z
Fick's law for mass
J i,x
Ci
  Dij
x
Fourier's law for heat
q
T
 k
A
x
Newton's law for
fluid momentum
τzx is shear stress
A is the surface area,
T is the temperature driving force,
q is the heat flow per unit time
k is the material's conductivity.
(Newtonian fluid)
Newtonian and non-Newtonian fluid
• In fluid mechanics, constitutive equations
provide the needed relations between the
shear stress and the fluid velocity
• Unlike a conservation relationship, which is
valid for all materials, a constitutive
relationship is not universal and applies to a
limited class of fluids
• Experimental measurements are needed to
derive constitutive relationship.
Non-Newtonian Rheology
• Rheology is the branch of mechanics
that studies the deformation of fluid
General expression:

app T , P,  x   zx
x
 For Newtonian fluid, the apparent viscosity is a constant
 For non-Newtonian fluid, the apparent viscosity (ηapp) changes
with the shear rate
 The apparent viscosity is a function of temperature (T), pressure
(P), and shear rate (γ)
Bingham plastic
• Bingham plastic does not flow until the applied
stress exceeds the yield stress (τ0) for fluid
• Below the yield stress, the shear rate and velocity
gradient vanish
x  0
 zx   0
• Above the yield stress, the relationship between
shear stress and shear rate is
 zx   0  zx   0  0 x
Power law fluids
• Power law fluid is a fluid for which the apparent
viscosity is a function of the shear rate raised to a
power
n 1
app  m  x
m and n depend on the particular fluid.
n=1: Newtonian fluid (m=µ)
n>1: Shear thickening or dilatant fluid
n<1: Shear thinning or pseudoplastic fluid
Relationship between shear rate,
shear stress and apparent viscosity
Comparison of convection and diffusion
Protein
Cell
Flask
10-8
 Length scale range over eight orders of
magnitude
 No single transport process can function
efficiently over these length scales
Lab fomenter
10-5
10-2
Fomenter
10-1
Factory
 At short distance, diffusion can be quite rapid
 As distance increases, the diffusion time
increases as the square of this distance
100 -101
102
Length scale: m
Comparison of convection and diffusion
• Evaluate the significance of diffusion and
convection
– Calculate Pectlet number
– Compare the time required for a molecule to be transported
by each process
Peclet number: represents the ratio of mass transfer by convection to
mass transfer by diffusion
Comparison of convection and diffusion
Peclet number
mass transport by convection  L2   u   uL 
Pe 

    


mass transport by diffusion  Dij   L   Dij 
 L2 
Diffusion time (td)
td  
 D  L is the characteristic length
 ij 
Convection time (tc)
L
tc   
u
u is the characteristic velocity
 When the Peclet number is much less than unity, diffusion is
more rapid than convection
 For short distance, convection is slower than diffusion
 For longer distances, diffusion is slower than convection
Comparison of convection and diffusion
Relative importance of diffusion and convection
MW, g mole-1
Dij (cm2s-1)
Diffusion
time (s)
Pe
Oxygen
32
2 ×10-4
5
0.05
Glucose
180
2 ×10-6
50
0.5
6,000
1 ×10-6
100
1.0
Antibody
150,000
6 ×10-7
167
1.67
Particles
Diameter (µm)
0.1
5 ×10-8
2,000
20
Bacterium
1
5 ×10-9
20,000
200
Cell
10
5 ×10-10
200,000
2,000
Molecule
Insulin
Virus
For L=100 µm, and if u=1 µm/s, the time for convection is always equal to L/u=100 s for all molecules
and particles
Comparison of convection and diffusion
 The distance at which transport by the two processes becomes equal is inversely
related to the diffusion coefficient
 For proteins (10-7), diffusion is an efficient process for dimensions on the order of the
size of a cell or smaller (10-3) .
 For small molecules (10-6) such as glucose, diffusion is efficient for distances on the
order of (10-2)
Comparison of transport and reaction
• Diffusion and convection often occur in conjunction
with specialized cellular transport processes and
chemical reactions
• It is important to quantify the relative contribution of
each of the different processes to the overall
transport process
• Evaluate the significance of transport and reaction
– Calculate Bi number
– Calculate Da number
Comparison of transport and reaction
Biot number is a dimensionless number that is a measure of
the relative resistances of each process
mass transfer across a cell layer
km L
Bi 

mass transfer by diffusion through tissue Deff
Km is the permeability of the cell layer
L is the distance over which diffusion in the tissues occurs
 If tissue is the major resistance, Bi is much greater than unity
 If membrane is the limiting resistance, Bi is much less than unity
Comparison of transport and reaction
Damkohler number (Da) represents the ratio of the mass transfer
time (tm) to the reaction time (tR)
k R tm
Da 

km tR
13
 u 
km  1.4674 Dij 
 D La 
 ij 
kR is the first-order heterogeneous rate constant
km is the length-averaged mass transfer coefficient
a is the gap height of the flow chamber
L is the width of the flow chamber
u is the average velocity
 For Da<<1, the rate of reaction is not limited by transport. The
observed rate of reaction is equal to the intrinsic rate of reaction,
and the reaction is said to be kinetically limited
 For Da>>1, this is mass transfer limited.
Transport within the cell