Cell Process
1. Cell and transport phenomena (3)
Liang Yu
Department of Biological Systems Engineering
Washington State University
02. 21. 2013
Main topics
Cell and transport phenomena
High performance computation
Metabolic reactions and C-13 validation
Enzyme and molecular simulation
General transport equations
• The general variable  : three velocity components,
enthalpy, temperature, and species concentration or
other conservative variables.
• There are significant commonalities between the
various equations. Using a general variable , the
conservative form of all fluid flow equations can
usefully be written in the following form:
  
 div  u   div grad    S
t
• Or, in words:
Rate of increase
of  of fluid
element
+
Net rate of flow
of  out of
fluid element
(convection)
=
Rate of increase
of  due to
diffusion
+
Rate of increase
of  due to
sources
Multi-scale model diagram
A single-cell-based model of
multicellular growth
Cell is the basic structural and functional unit of all living organisms. It is
critical to biofuel production. Because there is a relationship that the cells
begin to produce biofuel, which inhibits their growth, eventually killing the
entire population.
M. J. Dunlop, J. D. Keasling, A. Mukhopadhyay. A model for improving microbial biofuel production using a synthetic
feedback loop. Syst Synth Biol (2010) 4:95–104
Bioconvection
Self-swimming Algae: statistics of individual swimming and bioconvection
Bioconvection is a fluid dynamic phenomenon that originates in the
movement of microorganisms
The movement is caused by
Chemontaxis (attraction to a chemical)
Gyrotaxis (motion in response to gravity)
The bioconvection patterns created by different strains of bacteria are
often unique and can help to identify different organisms
Bioconvective plumes were thought to enhance bacterial growth by
increasing the overall amount of oxygen dissolved in the water
Modeling approaches
Continuum partial deferential equation (PDE) models
Explore the coupled dynamics of cellular
populations and biochemical substrates and
regulators of proliferation and cell death
This includes relatively simple models for nutrient
consumption in a tumor spheroid, and more
complex models with mechanical effects, multiple
interacting populations, and pattern formation on
growing domains
These approaches neglect the details of
individual cell growth and movement
Modeling approaches
Discrete models (cellular automaton and lattice-gas
automata)
Each cell is represented by a single automaton location,
and cell division and/or movement is determined by simple
rules
Application to the migration of contact inhibited cells, and
to cancer growth and its interaction with the immune
system
include models that incorporate extracellular diffusible
substances (e.g. nutrients), intracellular dynamics, such as
for the cell cycle, via systems of ODEs that model the
relevant regulatory networks, and coupling to other model
layers such as a vascular network or the extracellular matrix
which can mediate haptotaxis and invasion
Modeling approaches
Monte Carlo approach
Treats cells as elastic sticky spheres with a hard
center
Application to monolayer and spheroid cultures,
and liver regeneration
Does not explicitly include the fluid uptake that is
required for cell growth
Cell growth is dependent on the degree of cell
packing, with no restriction if cells are not
touching
Modeling approaches
Metabolic network simulation
Extreme Pathways
Elementary mode analysis
Minimal metabolic behaviors (MMBs)
Flux balance analysis
Dynamic simulation and parameter estimation
Only consider reactions, no transport inside and
outside of cell.
Modeling approaches
Molecular dynamics simulations (MDS)
On the molecular level, provide details in cell
Despite the increasing computational power of
workstations and supercomputers, simulation of
the whole cell at the molecular level remains
prohibitively expensive. For example a small
yeast cell contains 50 million proteins, still there
are more relevant molecules (DNA, RNA, lipids,
metabolites), and especially all the ions and water
molecules in the cell
Modeling approach in this study
Immersed boundary method
This model incorporates essential aspects of the
mechanical forces involved in growth and cell
division of individual cells, and in particular
explicitly includes the fluid sources required for
cellular volume changes
This approach provides the pressure and force
distribution within the tissue and can be used for
testing the influence of stress on cell proliferation
and death
Robert Dillon, Markus Owen, and Kevin Painter. A single-cell-based model of multicellular growth using the immersed
boundary method. AMS Contemporary Mathematics. 466:1-15,2008
Immersed boundary method (IB)
IB method is both a mathematical formulation and a numerical
scheme
Mathematical formulation employs a mixture of Eulerian and
Lagrangian variables
These are related by interaction equations in which the Dirac delta
function plays a prominent role
In the numerical scheme, the Eulerian variables are defined on a
fixed Cartesian mesh, and the Lagrangian variables are defined on
a curvilinear mesh that moves freely through the fixed Cartesian
mesh without being constrained to adapt to it in any way at all
The interaction equations of the numerical scheme involve a
smoothed approximation to the Dirac delta function, constructed
according to certain principles
Dirac delta function
The Dirac delta function, or δ function, is (informally) a
generalized function on the real number line that is zero
everywhere except at zero, with an integral of one over
the entire real line.
, x  0
  x  
 0, x  0

   x  dx  1

Peskin’s IB Method
Introduced in the 70s
to simulate the flow in
the human heart
Navier-Stokes
equations are solved
on a Cartesian grid.
Heart walls are
modeled as elastic
membranes. The
interaction is modeled
using a source term
added to the governing
equations
Peskin’s IB Method
Eulerian fluid governing equations
Lagrangian fiber tracking
Coupling term
Key ingredients:
and
Mathematical model
Each cell is modeled as a viscous fluid
with additional elastic forces representing
the cell membrane
The material properties of the cell wall are
modeled via a network of linear elastic
springs
During the growth process, additional fluid
is introduced into the cell's interior via
discrete fluid channels located around the
circumference of the cell
Each channel is modeled as a discrete
source and sink
In animal cells a contractile ring of actin
and myosin filaments contracts during
cleavage to form the two daughter cells
The elastic links between neighboring cells
mediate cell-cell adhesion and can also
maintain a minimal separation distance
between cells.
(a)
(b)
(a) Schematic of model cell. The cell wall is
represented as a mesh of linear elastic
forces. The transport of fluid from the
exterior to the interior is facilitated via
discrete channels modeled as source
(o) and sink (+) pairs. The contractile
force links for cell division extend
across the cell.
(b) Simulation detail of cell to cell link
structure.
Mathematical equations
Continuity equation
Cell growth
u  S  c, x, t 
Navier-Stokes equation
u
1
 2


   u   u  p     u  S   F
t
3


F is the force density that the cells and links exert on the fluid
F =  Fcell (i )   Flink ( j )   Fcontractile ( k )
i
j
k
Mathematical equations
Cell model
Fcell (i ) =  f cell (i )  r , s, t    x  X i  r , s, t   drds
A Lagrangian force per unit area fcell(i)(r, s, t) is defined at each point on the ring. X is finite
thickness of cell membrane. This immersed boundary force is transmitted directly to the
fluid and gives a contribution to Eulerian Fcell(i).
X i  r , s, t 
 u  X i  r , s, t  , t    u  x, t   x  X i  r , s, t   dx
t
The cells Xi move at the local fluid velocity
f qr  Scell
X
r
 X q  DL

Xr  Xq
Xr  Xq
The force fqr at the immersed boundary point Xq due to the elastic link with the immersed
boundary point Xr is given by Hooke's Law. Scell is the spring force constant and DL is the
spring resting length
Mathematical equations
Cell growth
S  c, x, t     Sij  Sij 
ij
Ssij is the contribution of the jth source (s = +) or sink (s = -) for the ith cell and has the form.
Sijs  x, t    Sij  x  X ijs 
Sij  K 0Cij
Sij is the growth rate constant, K0 is the uptake rate constant and Cij is the local nutrient
concentration.
Substrate kinetics and transport
0  DC 2C  f 
Simulated results for tumor cell
Simulation of cell division at the beginning (a), middle (b) , and end
(c) of the division process. The two daughter cells are shown in (d)
Simulated results for tumor cell
Cell growth simulation with chemistry
Simulated results for tumor cell
Top Row: Numerical Simulations of cell spheroid with nutrient uptake rate (a) 0.001
(785, 56days) (b) 0.01 (502, 183days) (c) uniform growth (793, 129days). The number
of cells is shown in parentheses. Bottom Row: Scatter plots. The dots indicate the
time (x-axis) and distance (y-axis) from the cell cluster centroid .
Simulated results for tumor cell
Simulations with necrosis at threshold levels cmin (at times) (a) 0.0 (166 days) (b)
0.00001 (166 days) (c) 0.0015 (180 days) (d) 0.05 (171 days). Here, the nutrient
uptake rate is set to 0.01.
Simulated results for tumor cell
Tufting (top) and solid (bottom) patterns in DCIS(ductal
carcinoma in situ)
Simulated results for tumor cell
Solid pattern in DCIS (ductal carcinoma in situ)
Summary of model complexity.
Holmes WR, Edelstein-Keshet L (2012) A Comparison of Computational Models for Eukaryotic Cell Shape and Motility. PLoS
Comput Biol 8(12): e1002793. doi:10.1371/journal.pcbi.1002793
http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002793
Finite volume 2-D simulations.
Holmes WR, Edelstein-Keshet L (2012) A Comparison of Computational Models for Eukaryotic Cell Shape and Motility. PLoS
Comput Biol 8(12): e1002793. doi:10.1371/journal.pcbi.1002793
http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002793
Finite element 2-D simulations.
Holmes WR, Edelstein-Keshet L (2012) A Comparison of Computational Models for Eukaryotic Cell Shape and Motility. PLoS
Comput Biol 8(12): e1002793. doi:10.1371/journal.pcbi.1002793
http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002793
Keratocyte motility by LSM (Level set method)
Holmes WR, Edelstein-Keshet L (2012) A Comparison of Computational Models for Eukaryotic Cell Shape and Motility. PLoS
Comput Biol 8(12): e1002793. doi:10.1371/journal.pcbi.1002793
http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002793
Pressure-driven flow through a
Cylindrical tube
Laminar flow of a Newtonian fluid through a cylinder of radius R
and momentum balance on a differential volume r∆θ∆r∆z
Flow is steady, and fully developed
Pressure-driven flow through a
Cylindrical tube
Such flows arise in many biomedical applications,
such as flow in ultrafiltration and dialysis units,
bioreactors needles, infusion systems and
capillary tube viscometers.
Since there is no net momentum flow and the flow
is steady, the sum of all forces must equal zero.
The only forces arising are those due to pressure
and shear stress.
A momentum balance in the z direction yields
p
z


 p z z  rr   r  r  rz r r  r  rz r z  0
Pressure-driven flow through a
Cylindrical tube
Dividing by r∆θ∆r∆z and taking the limit as each goes to zero
results in the ordinary differential equation
dp d  r rz 

dz
rdr
The pressure changes only in the z direction (i.e., dp/dz=f(z)) and
the shear stress changes in the r direction (i.e., d(rτ)/dr=g(r))
f ( z)  g (r)
Integrate f(z) to yield
p  C1z  C2
Pressure-driven flow through a
Cylindrical tube
The pressure can be specified at two locations, away from both the
entrance and exit. Thus, at z=z0, p=p0, and at z=zL, p=pL. Defining
∆p=p0-pL and L=zL-z0 to remove C1 and C2
p
p  p0 
 x0  x 
L
Integrate τ to yield
d  r rz 
p

rdr
L
pr C3
 rz  

2L
r
Since τ must be finite at r=0, C3 must equal zero.
Newton’s law of viscosity
 rz  
duz
dr
Pressure-driven flow through a
Cylindrical tube
Substitute into Newton’s law of viscosity
duz
pr

dr
2 L
After integrating this equation
pr 2
uz  
 C4
4 L
Apply the no-slip boundary condition at r=R
pR 2
C4 
4 L
pR 2 
r2 
uz 
1 2 

4 L  R 
Pressure-driven flow through a
Cylindrical tube
The velocity is a maximum at y=0
umax
pR 2

4 L

r2 
uz  umax  1  2 
 R 
The volumetric flow rate is the integral of the velocity over the
cross-sectional area
Q
R
0

2
0
uz rd dr  umax 2 
R
0

r2 
p R 4
1  R 2  rdr  8 L


Pressure-driven flow through a
Cylindrical tube
Analytic solution
 rz  
pr
2L
pR 2 
r2 
uz 
1 2 

4 L  R 
Numerical solution
d  r rz 
p

rdr
L
 rz  
duz
dr
Pressure-driven flow through a
Cylindrical tube
Use Matlab to provide analytic solution and numerical solution
∆ p= 1(Pa)
R=0.05(m)
L=xL-x0=1.5(m)
µ= 1.0 x 10-3 (Pa s, N s/m2)
function NewtonianFluidFlowCylindrical
% Pressure-driven flow through a cylindrical tube
% Solve momentum equation to obtain shear stress and velocity
distribution
% Laminar Flow in a Horizontal Pipe (Newtonian Fluid)
%
clear all
clc
global deltaP L mu R
deltaP = 1;
L = 1.5;
mu = 1.0e-3;
R = 0.05;
a = 0;
b = R;
% Sovle the problem of ODE-BVP
% initialize of solution with a guess of y1(r)=0,y2(r)=0,y3(r)=0
solinit = bvpinit(linspace(a,b,100),[0 0 0]);
sol = bvp4c(@ODEs,@BCfun,solinit);
Pressure-driven flow through a
Cylindrical tube
% Analysis results
r = sol.x;
TauAnal = (deltaP/(2*L))*r;
uAnal = (deltaP*R^2/(4*mu*L))*(1-(r/R).^2);
%umAnal = deltaP*R^2/(8*mu*L);
% Plot
% Shear stress
tau = sol.y(1,2:end)./sol.x(2:end);
tau = [0 tau];
plot(r,tau,'b-',r,TauAnal,'r-.')
xlabel('Pipe redius r£¬m')
ylabel('Shear stress£¬kg/(m s^2)')
legend('Numerical results','Analysis results')
figure
% Velocity distribution
plot(r,sol.y(2,:),'b-',r,uAnal,'r-.')
xlabel('Pipe redius r£¬m')
ylabel('Velocity£¬m/s')
legend('Numerical results','Analysis results')
% Average velocity
fprintf('\tAverage velocity: um = %.4f',sol.y(3,end))
Pressure-driven flow through a
Cylindrical tube
function dydr = ODEs(r,y)
global deltaP L mu R
rTau = y(1);
u = y(2);
um = y(3);
drTaudr = deltaP*r/L;
if r > 0
% Avoid zero
Tau = rTau/r;
else
Tau = 0;
end
dudr = -Tau/mu;
dumdr = u*2*r/R^2;
dydr = [drTaudr; dudr; dumdr];
% -----------------------------------------------------------------function bc = BCfun(ya,yb)
bc = [ya(1); yb(2); ya(3)];
Pressure-driven flow through a
Cylindrical tube