Introduction to Mathematical Biology · 1. Modelling biological and biochemical process using ODEs Osvaldo Chara Monday 13 and Tuesday 14, 9:00 to 10:30 h. 2. Modelling biological

Introduction to Mathematical Biology

Osvaldo Chara Technische Universität Dresden, Germany

[email protected]

http://imc.zih.tu-dresden.de/imc/

Osvaldo Chara - ESSIM - Dresden 2012 - Tuesday

1

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


2

1. Modelling biological and biochemical process using ODEs Osvaldo Chara

Tuesday 14

9:00 to 10:30 h.


3

Yeah, I am still here


4

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


5

electrolytes

Transport of electrolytes: Nernst equation


6

electrolytes

zoom



7

one molecule of electrolyte

zoom



8

let us see…

frictional force

electrical force



9

The Flux cvJ d

Then ck

qEJ

Homogeneous concentration!



10

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



11

In equilibrium

Then

0J

Then

cz

c

Tk

qE

B

1

d

d

x

x

cx

Tk

qExx

B

dd

lndd

00



12

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!



13

xc

c

q

TkV B 0

ln

The Nernst equation!

I like it so much!



14

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


15

a

VzFUc

dx

dcURTJ

ac

c

a

Ja

VzFcU

dcRTUdx

00

Then …

Integrating …



16

RT

VzF

RT

VzF

ie

e

ecc

a

VzFUJ

1

b

e

s

e cc b

i

s

i cc But remember …

Therefore …



17

RT

VzF

RT

VzF

ie

e

ecc

a

VzFUJ

1


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.


18

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


19

An example: the dynamics of the extracellular ATP (ATPe)


20

ATPi

ATPe ADPe

k1

k2 k3



21

eei

e ATPkATPATPkt

ATP21

d

d

ee

e ADPkATPkt

ADP32

d

d

0

d

d

t

ATPi



22

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:


23

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

[email protected]

http://imc.zih.tu-dresden.de/imc/

Thanks a lot!


24

Documents

Introduction to Mathematical Biology · 1. Modelling biological and biochemical process using ODEs Osvaldo Chara Monday 13 and Tuesday 14, 9:00 to 10:30 h. 2. Modelling biological