introduction to mathematical biology · 1. modelling biological and biochemical process using odes...
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Introduction to Mathematical Biology
Osvaldo Chara Technische Universität Dresden, Germany
http://imc.zih.tu-dresden.de/imc/
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1. Modelling biological and biochemical process using ODEs Osvaldo Chara
Monday 13 and Tuesday 14, 9:00 to 10:30 h.
2. Modelling biological and biochemical process using PDEs Lutz Brusch
Wednsesday 15 and Thursday 16, 13:30 to 15:00 h.
Schedule
Osvaldo Chara - ESSIM - Dresden 2012 - Tuesday
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1. Modelling biological and biochemical process using ODEs Osvaldo Chara
Tuesday 14
9:00 to 10:30 h.
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Yeah, I am still here
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transport of water and non-electrolyte solutes
j
ej
ij
jwf cv
mAvP
t
v
d
d
Kedem & Katchalsky, 1958. Thermodynamic analysis of the permeability of biological membranes to non-electrolytes. Biochim. Biophys. Acta 27, 229-246. Kedem & Katchalsky, 1961. A physical interpretation of the phenomenological coefficients of membrane permeability. J. Gen. Physiol. 45: 143-179. Katchalsky & Curran, 1965. Non-equilibrium Thermodynamics in Biophysics. Harvard University Press, Cambridge, MA, USA. Kleinhans, 1998. Membrane permeability modeling: Kedem-Katchalsky vs a two-parameter formalism. Cryobiology. 37(4):271-89.
ej
ij
j
jc
v
mAP
t
m
d
d
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electrolytes
Transport of electrolytes: Nernst equation
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electrolytes
zoom
Transport of electrolytes: Nernst equation
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one molecule of electrolyte
zoom
Transport of electrolytes: Nernst equation
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let us see…
frictional force
electrical force
Transport of electrolytes: Nernst equation
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The Flux cvJ d
Then ck
qEJ
Homogeneous concentration!
Transport of electrolytes: Nernst equation
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The Flux
Thanks to Einstein we know that
x
cDc
k
qEJ
d
d
Then
D
Tkk B
Dx
c
Tk
qEcJ
B
d
d
This is the Nernst-Planck equation
Transport of electrolytes: Nernst equation
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In equilibrium
Then
0J
Then
cz
c
Tk
qE
B
1
d
d
x
x
cx
Tk
qExx
B
dd
lndd
00
Transport of electrolytes: Nernst equation
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Then
0
lnd0
c
xcxE
Tk
qx
B
x
VE
d
d
x
xEV0
d
xc
cV
Tk
q
B
0ln
But…
xc
c
q
TkV B 0
ln
The Nernst equation!
Transport of electrolytes: Nernst equation
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xc
c
q
TkV B 0
ln
The Nernst equation!
I like it so much!
Transport of electrolytes: Nernst equation
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x
VzFUc
x
cURTJ
d
d
d
d
a
V
x
V
d
d
Assuming …
This is again the Nernst-Planck equation
And …
bs cc
Transport of electrolytes: Goldman equation
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a
VzFUc
dx
dcURTJ
ac
c
a
Ja
VzFcU
dcRTUdx
00
Then …
Integrating …
Transport of electrolytes: Goldman equation
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RT
VzF
RT
VzF
ie
e
ecc
a
VzFUJ
1
b
e
s
e cc b
i
s
i cc But remember …
Therefore …
Transport of electrolytes: Goldman equation
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RT
VzF
RT
VzF
ie
e
ecc
a
VzFUJ
1
Transport of electrolytes: Goldman equation
Goldman. 1943. Potential, Impedance, and Rectification in membranes. J Gen Physiol 27:37–60.
Hodgkin & Huxley, 1952. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. I I7, 500-544.
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RT
VFz
RT
VFz
ijejjjj
jj
j
e
ecc
a
VFzUJ
1
j
ej
ij
jwf cv
mAvP
t
v
d
d
jj
jJr
t
m
d
d
Adding all up: the intracellular dynamics
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An example: the dynamics of the extracellular ATP (ATPe)
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ATPi
ATPe ADPe
k1
k2 k3
An example: the dynamics of the extracellular ATP (ATPe)
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eei
e ATPkATPATPkt
ATP21
d
d
ee
e ADPkATPkt
ADP32
d
d
0
d
d
t
ATPi
An example: the dynamics of the extracellular ATP (ATPe)
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eei ATPkATPATPk 210
ee ADPkATPk 320
0
d
d
d
d
t
ADP
t
ATP ee
21
1
kk
ATPkATP i
e
213
12
kkk
ATPkkADP i
e
Some references of our work:
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Chara O, Espelt MV, Krumschnabel G, Schwarzbaum PJ. 2011. Regulatory volume decrease and P receptor signaling in fish cells: mechanisms, physiology, and modeling approaches. J Exp Zool A Ecol Genet Physiol. 315(4):175-202. Chara O, Pafundo DE, Schwarzbaum PJ. 2010. Negative feedback of extracellular ADP on ATP release in goldfish hepatocytes: a theoretical study. J Theor Biol. 264(4):1147-58. Chara O, Pafundo DE, Schwarzbaum PJ. 2009. Kinetics of extracellular ATP from goldfish hepatocytes: a lesson from mathematical modeling. Bull Math Biol. 71(5):1025-47. Alleva K, Chara O, Sutka MR, Amodeo G. 2009. Analysis of the source of heterogeneity in the osmotic response of plant membrane vesicles. Eur Biophys J. 38(2):175-84. Pafundo DE, Chara O, Faillace MP, Krumschnabel G, Schwarzbaum PJ. 2008. Kinetics of ATP release and cell volume regulation of hyposmotically challenged goldfish hepatocytes. Am J Physiol Regul Integr Comp Physiol. 294(1):R220-33.
Osvaldo Chara Technische Universität Dresden, Germany
http://imc.zih.tu-dresden.de/imc/
Thanks a lot!
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