Introduction to Mathematical Biology
Osvaldo Chara Technische Universität Dresden, Germany
http://imc.zih.tu-dresden.de/imc/
Osvaldo Chara - SSIM - Dresden 2012 - Monday
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
Osvaldo Chara - SSIM - Dresden 2012 - Monday
2
1. Modelling biological and biochemical process using ODEs Osvaldo Chara
Monday 13
9:00 to 10:30 h.
Osvaldo Chara - SSIM - Dresden 2012 - Monday
3
yxt
x
d
d
xyt
y
d
d
the Lotka-Volterra model By the way, I am Vito Volterra
rxt
x
d
d
the Malthus model
xxrt
x
d
d
the logistic model
Osvaldo Chara - SSIM - Dresden 2012 - Monday
4
Rashevsky, N. (Ed.), 1962. Physicomathematical Aspects of Biology: Proceedings of the International School of Physics “Enrico Fermi” XVI Course. Academic Press, New York.
Osvaldo Chara - SSIM - Dresden 2012 - Monday
5
Rashevsky, N. (Ed.), 1962. Physicomathematical Aspects of Biology: Proceedings of the International School of Physics “Enrico Fermi” XVI Course. Academic Press, New York.
Osvaldo Chara - SSIM - Dresden 2012 - Monday
6
Yeah, I am Nicolas Rashevsky
Osvaldo Chara - SSIM - Dresden 2012 - Monday
7
molecules cells tissues organs organisms populations
Which is the spatial scale of the phenomenon under study?
Osvaldo Chara - SSIM - Dresden 2012 - Monday
8
molecules cells tissues organs organisms populations
Which is the spatial scale of the phenomenon under study?
Osvaldo Chara - SSIM - Dresden 2012 - Monday
9
Osvaldo Chara - SSIM - Dresden 2012 - Monday
10
modelling cellular processes
reactions transport
Osvaldo Chara - SSIM - Dresden 2012 - Monday
11
modelling cellular processes
reactions transport
Osvaldo Chara - SSIM - Dresden 2012 - Monday
12
A + B C + D
reactants products
chemical reaction
Osvaldo Chara - SSIM - Dresden 2012 - Monday
13
A + B C + D
reactants products
chemical reaction
1. stechiometry
2. directionality / reversibility
Osvaldo Chara - SSIM - Dresden 2012 - Monday
14
A + B C + B
catalyst product
catalysis
substrate catalyst
Osvaldo Chara - SSIM - Dresden 2012 - Monday
15
A + B C + B
catalyst product
catalysis
substrate catalyst
Osvaldo Chara - SSIM - Dresden 2012 - Monday
16
A C
catalyst product
catalysis
substrate
B
Osvaldo Chara - SSIM - Dresden 2012 - Monday
17
2H2O2 2H20 + O2
Catalase
just an example
Osvaldo Chara - SSIM - Dresden 2012 - Monday
18
modelling cellular processes
reactions transport
Osvaldo Chara - SSIM - Dresden 2012 - Monday
19
modelling cellular processes
reactions transport
Osvaldo Chara - SSIM - Dresden 2012 - Monday
20
Apical membrane
Basolateral membrane
Paracelular path Transcelular path
lipid bilayer AQP cotransporter
Osvaldo Chara - SSIM - Dresden 2012 - Monday
21
Apical membrane
Basolateral membrane
Paracelular path Transcelular path
lipid bilayer AQP cotransporter
AQPs!
Osvaldo Chara - SSIM - Dresden 2012 - Monday
22
j
ejijjwfv ccAvPJ
transport of water: osmosis
Osvaldo Chara - SSIM - Dresden 2012 - Monday
23
j
ejijjwfv ccAvPJ
transport of water: osmosis
v
mc
ij
ij vJt
v
d
d
Osvaldo Chara - SSIM - Dresden 2012 - Monday
24
j
ejijjwfv ccAvPJ
transport of water: osmosis
v
mc
ij
ij vJt
v
d
d
j
ej
ij
jwf cv
mAvP
t
v
d
d
Pickard WF. 2008. Modelling the swelling assay for aquaporin expression. J Math Biol 57:883–903. Chara et al., 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.
Osvaldo Chara - SSIM - Dresden 2012 - Monday
25
The cup of coffee of SSIM 2012
coffee
sugar
Osvaldo Chara - SSIM - Dresden 2012 - Monday
26
The cup of coffee of SSIM 2012
coffee
sugar
Osvaldo Chara - SSIM - Dresden 2012 - Monday
27
The cup of coffee of SSIM 2012
coffee
sugar
Osvaldo Chara - SSIM - Dresden 2012 - Monday
28
The cup of coffee of SSIM 2012
coffee
sugar
Osvaldo Chara - SSIM - Dresden 2012 - Monday
29
a phenomenological explanation: the Fick's law
Imaginary membrane of area A and length x
Js
Cx
ADJ s
s
Osvaldo Chara - SSIM - Dresden 2012 - Monday
30
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
Osvaldo Chara - SSIM - Dresden 2012 - Monday
31