Transcript Matematik i biologi og farmaceutisk industri. Årskursus i matematik, kemi og fysik, Rosborg Gymnasium, Vejle, 24/10, 2008 Mads Peter Sørensen DTU Matematik, Kgs.

Matematik i biologi og farmaceutisk industri.
Årskursus i matematik, kemi og fysik, Rosborg Gymnasium, Vejle, 24/10, 2008
Mads Peter Sørensen
DTU Matematik, Kgs. Lyngby
Indhold:
1)
Udvikling af medicin og matematisk modellering.
2)
Blodkoagulation.
3)
Insulinproducerende beta celler.
4)
Sammenfatning.
Samarbejdspartnere.
1)
Nina Marianne Andersen, DTU Matematik og Novo Nordisk
2)
Steen Ingwersen, Biomodellering, Novo Nordisk.
3)
Ole Hvilsted Olsen, Hæmostasis biokemi, Novo Nordisk.
4)
Morten Gram Pedersen, Department of Information Engineering, University of
Padova, Italy.
5)
Oleg V. Aslanidi, Institute of Cell Biophysics RAS, Pushchino, Moscow, Russia.
6)
Oleg A. Mornev, Institute of Theoretical and Experimental Biophysics RAS,
Pushchino, Moscow, Russia.
7)
Ole Skyggebjerg, Novo Nordisk.
8)
Per Arkhammar og Ole Thastrup, BioImage a/s, Søborg.
9)
Alwyn C. Scott, DTU Informatik og University of Arizona, Tucson AZ, USA.
10)
Peter L. Christiansen, DTU Fysik og DTU Informatik.
11)
Knut Conradsen, DTU Informatik
Sponsorer: Modelling, Estimation and Control of Biotechnological Systems (MECOBS).
EU Network of Excellence BioSim.
Udviklingsomkostninger for ny medicin.
Ref.: Erik Mosekilde, Ingeniøren 10. oktober, side 9, (2008).
EU Network of Excellence BioSim. http://biosim-network.eu
Udviklingsprocessen for ny medicin.
1) Opdagelse.
2) Prækliniske forsøg.
Ide, hypotese, forskning.
Udviklingsfase. Dyreforsøg.
Dyremodeller. Dyreforsøg.
Protokol for sikkerhed og
effektivitet.
Mekanisme og potentiel
giftpåvirkning af organer.
4) Godkendelse.
3) Kliniske forsøg.
Godkendelse fra regulerende
myndigheder.
Test på mennesker.
Test for sikkerhed og
effektivitet.
Regulerende myndigheder.
Godkendelse af
medikamentet.
Marketing autorisation.
Sikker og effektiv medicin.
>50% af udviklings tiden.
1 ud af 10-15 medikamenter
overlever til fase 3
5) Kontrol.
Lægemiddelovervågning
Matematisk modellering som et redskab i
udviklingen af ny medicin
Udviklingsomkostningerne for et nyt medikament ligger typisk mellem 1 og
7 milliarder kr.
Udviklingstid: 10 – 15 år.
Anvendelse af moderne modellerings og computer simuleringsværktøjer til
udvikling af ny medicin. Kompleksitet.
Mere rationel og hurtigere udviklings proces med færre
økonomiske omkostninger.
Forbedret behandling af patienter. Bedre, mere sikker og mere
individuel behandling.
Reduktion i anvendelse af dyre eksperimenter.
Computer model af menneske.
Disorders of Coagulation
Hypocoagulation:
Hypercoagulation:
Hemophilia A
Cardiovascular diseases:
Hemophilia B
Arthroscleroses
Others
Emboli and thrombi development
Cartoon of the blood
coagulation pathway.
Ref:
http://www.ambion.com/tool
s/pathway/pathway.php?pat
hway=Blood%20Coagulation
%20Cascade
Perfusions eksperiment og modellering
Perfusions kammer
Aktive thrombocyter (Ta) binder til et
collagen coated låg. vWF.
Glaslåg coated med collagen
Faktor X i fluid
fase
X
Thrombocyter (blodplader), røde og hvide
blod celler.
Faktor VIIa I fluid
fase
Rekonstrueret blod.
VIIa
Indhold: Thrombocyter (T), Erythrocyter.
[T] = 14 nM (70,000 blodplader / μ litre blood)
Enzym kinetic
k1
Reaktions skema:
k2
SE C EP
k 1
Reaktions ligningerne:
ds
 k 1c  k1se
dt
dc
 k1se  (k 1  k 2 )c
dt
Bemærk at:
e  c  e0
de
 (k 1  k 2 )c  k1se
dt
dp
 k2c
dt
Enzym kinetic
Skalering:
s

s0
Matematisk model:
c
x
e0
k1  k2

k1s0
e0

s0
k1

k1s0
d
   x(   )
d
dx

   x(   )
d
Kvasistationær tilstand:
x   /(   )
d
(   )

d
 
Ref.: J. Keener and J. Sneyd, Mathematical Physiology, Springer, New York, (1998).
M.G. Pedersen, A.M. Bersani and E. Bersani, Jour. of Math. Chem. 43(4), pp1318-1344, (2008).
Konkurrerende inhibitor (hæmningsstof)
k1
Reaktions skema:
k2
S  E  C1  E  P
k 1
k3
E  I  C2
k 3
Inklusion af flow og diffusion:
s

 Ds s  (v s )  k 1c1  k1se
t
Diffusionskonstant:
Ds
Reaktionsskema ved rand:
k4
PB 
BP
k
4
Konvektions flow hastighed:
Bindingssites på rand:
B
B  BP  B0
db
  k 4bp  k  4 (b0  b)
dt

v
To dimensionalt eksempel med flow, diffusion og bindingssites på
randen
P
y
x
Bindingssites på randen:
Cartoon model of the perfusion experiment
Unactivated Platelet
Activated Platelet
IIa
II
IIa
IIa
VIIa
X
Xa
V
Activated Platelet
Va
Va:Xa
Reaction schemes, one example.
II  Ta  II Ta
Ta10
Factor II (prothrombin): II
F 10
Factor IIa (thrombin): IIa
R16
II Ta  XaTa  XaTa  IIa
Prothrombinase complex: Xa_Va_Ta
S3
II Ta  XaVaTa  XaVaTa  IIa
A total of 17 equations.
dII Ta
  R16  XaTa  II Ta  S 3  XaVaTa  II Ta 
dt
Ta10  II  Ta  F10  II Ta
R16  7500 M 1s 1
Reaction rates:
1 1
S 3  10 M s
7
F 10  1s
1
Ta10  43000 M
1 1
s
Ref: P.M. Didriksen, Modelling hemostasis - a biosimulation project,
internal report, Dept. 252 Biomodelling, Novo Nordisk
Numerical results.
Initial conditions: FVIIa = 50 nM FX = 170 nM T = 14 nM sTa = 0.1*14 nM FII = 0.3 nM
IIa
T
VIIa
Ta
Reaction diffusion model with convection
Reaction scheme for T, Ta and IIa.
T6
T7
T  IIa  sTa  Ta  IIa
Corresponding model equations in the space Ω.
dT
 T6  T  IIa  DT   2T  (v( y )  T )
dt
dIIa
 T7  sTa  T6  T  IIa  DIIa  2 IIa  (v( y )  IIa)
dt
dsTa
 T6  T  IIa  T7  Ta  IIa  DsTa   2 sTa  (v( y )  sTa )
dt
dTa
 T7  sTa  DTa   2Ta  (v( y )  Ta)
dt
Poiseuille’s flow v( y)  ay(1  y / H )
Boundary conditions and parameters
Boundary condition x=0
IIa  1.2 106 nM  f1 ( y )
T  14 109 nM  f 2 ( y )
Ta  0
sTa  0
Boundary condition x=l: Outflow boundary conditions.
Top and bottom boundary condition: No flow crossing.
Ref.: M. Anand, K. Rajagopal, K.R. Rajagopal. A Model Incorporating some of the Mechanical and
Biochemical Factors Underlying Clot Formation and Dissolution in Flowing Blood. Journal of
Theoretical Medicine, 5: 183-218, 2003.
Numerical results.
T
T-IIa
IIa
Ta
Time = 0.6 sec.
Numerical results.
T
IIa
T-IIa
Ta
Time = 5 sec.
Numerical results.
T
T-IIa
IIa
Ta
Time = 10 sec.
Future work: Boundary attachment of Ta
Reaction schemes on 
k2
Ta  TaB
k3
T  TaB
k5
k4
II  TaB  C2  TaB  IIa
Corresponding model equations on.

dII
dIIa
 k 4  II  TaB
 k5  C2
dt
dt
dTaB
 k 2  Ta  k 4  II  TaB  k5  C2  k3  T
dt
dC2
 k 4  II  TaB  k5  C2
dt
Including pro-coagulant and anti-coagulant thrombin
Ref.: V.I. Zarnitsina et al, Dynamics of spatially nonuniform patterning in the model of blood coagulation,
Chaos 11(1), pp57-70, 2001.
E.A. Ermakova et al, Blood coagulation and propagation of autowaves in flow, Pathophysiology og
Haemostasis and Thrombosis, 34, pp135-142, 2005.
Model consisting of 11 PDEs in 2+1 D, including diffusion
Sammenfatning og fremtidig arbejde
1. Modellering af perfusionseksperiment for blodkoagulation.
2. Reduceret PDL model, som inkludere blod flow og
diffusion.
3. Modellering af vedhæftning af aktive thrombocyter på
collagen coated rand.
4. Fuld PDL model.
5. Model af in vivo blod koagulation.
Synthesis and secretion of insulin
Pre-proinsulin
Transcription
Insulin
Golgi complex
packed in granules
Proinsulin
Endoplasmatic reticulum
B
Exocytosis of insulin
caused by increased Ca concentration
The β-cell
Ion channel gates for Ca and K
B
Mathematical model for single cell dynamics
The modified Hodgkin-Huxley model for a single β-cell
C
dv
  I Ca  I K  I s  I K ( ATP )  I K ( Ca )  I CRAC
dt
Ion currents due to the ion-gates
I Ca  g Ca m (v(t )  vCa )
I s  g s s (t )(v(t )  vs )
I K  g K n(t )(v(t )  vK )
I K ( ATP )  g K ( ATP ) (v(t )  vK )
Ref.: A. Sherman, (Eds. Othmar et al), Case studies in mathematical modelling, ecology
physiology and cell biology, Prentice Hall (1996), pp.199-217.
Ref.: Fall, Marland, Wagner, Tyson, Computational Cell Biology, Springer, (2002).
Mathematical model for single cell dynamics
The gating variables
dn n  n(t )

dt
n
1
x 
 vx  v 

1  exp 
 sx 
ds s  s(t )

dt
s
k
C O
k
Ref.: Fall, Marland, Wagner, Tyson, Computational Cell Biology, Springer, (2002).
Ref.: Fall, Marland, Wagner, Tyson, Computational Cell Biology, Springer, (2002).
Dynamics and bifurcations
Ref.: E.M. Izhikevich, Neural excitability spiking and bursting, Int. Jour. of Bifurcation and Chaos,
p1171 (2000).
Dynamics and bifurcations
Simple polynomial model
dx
 y  x 3  3x 2  I  z
dt
dy
 1  5x 2  y
dt
dz
 r s ( x  x1 )  z 
dt
Parameters
x1  (1  5) / 2
r  0.001
s4
Ref.: J. Keener and J. Sneyd, Mathematical Physiology, Springer, New York, (1998).
Sketch of the homoclinic bifurcation
z  z cr
z  z cr
z  z cr
Mathematical model for single cell dynamics
Topologically equivalent and simplified models. Polynomial model
with Gaussian noise term on the gating variable.
du
 f (u )  w  z
dt
dw
 g (u )  w  (t )
dt
Voltage across the cell membrane:
Gating variable:
w  w(t )
Gaussian gate noise term:
(t )
dz
  (h(u )  z )
dt
u  u(t )
Slow gate variable:
where
z  z (t )
 (t )  0
 (t )(0)   (t )
Ref.: M. Panarowski, SIAM J. Appl. Math., 54 pp.814-832, (1994).
Ref.: A. Sherman, (Eds. Othmar et al), Case studies in mathematical modelling, ecology physiology and
cell biology, Prentice Hall (1996), pp.199-217.
The influence of noise on the beta-cell bursting phenomenon.
 0
  0 .1
  0 .3
Ref.: M.G. Pedersen and M.P. Sørensen, SIAM J. Appl. Math., 67(2), pp.530-542, (2007).
Mathematical model for coupled β-cells
Gap junctions between
neighbouring cells
Coupling to nearest neighbours.
Coupling constant:
g ij
dvi
C
  I Ca  I K  I s  I K ( ATP )   g ij (vi  v j )
dt
j
Ref.: A. Sherman, (Eds. Othmar et al), Case studies in mathematical modelling, ecology physiology and
cell biology, Prentice Hall (1996), pp.199-217.
Coupled β-cells
Image analysis experiments of in vitro islets of Langerhans
Experiments on Islets of Langerhans
g ij
The gating variables
Calcium current:
I Ca  g Ca m (vi )(vi  vCa )
Potassium current:
I K  g K n(t )(vi  vK )
ATP regulated potassium current:
I K ( ATP )  g K ( ATP ) (vi  vK )
I s  g s s (t )(vi  vK )
Slow ion current:
The gating variables obey.
dn n (v)  n

dt
n
ds s (v)  s

dt
s
x  m, n, s
1
x  x (vi ) 
1  exp( (vi  vx ) / sx )
Glycose gradients through Islets of Langerhans
Ref.: J.V. Rocheleau, et al, Microfluidic glycose stimulations … , PNAS, vol 101 (35), p12899 (2004).
Glycose gradients through Islets of Langerhans. Model.
Continuous spiking for:
Bursting for:
Silence for:
Coupling constant:
Note that
i  43
g K ( ATP )  90 pS
90 pS  g K ( ATP )  162 pS
162 pS  g K ( ATP )
g K ( ATP )  120 pS  (i  1) 1 pS
corresponds to
g K ( ATP )  162 pS
i  1,2,..., N
Wave blocking
Units
t phys  kt t
kt  c / g Ca  5.3ms
ku  sm  12mV
u phys  ku u
Glycose gradients through Islets of
Langerhans
g ij  50 pS
PDE model. Fisher’s equation
Continuum limit of
dvi
 F (vi , si )  g c (vi 1  2vi  vi 1 )
dt
ut  f (u; a )  u xx
Is approximated by the Fisher’s equation
where
Simple kink
solution
f (u; a)  u(u  a)(1  u)
1
u 0 ( x, t ) 
 x  x0  vt 
1  exp 

2


Velocity:
v  (1  2a ) / 2
Ref.: O.V. Aslanidi et.al. Biophys. Jour. 80, pp 1195-1209, (2001).
Numerical simulations and comparison to analytic result
Sammenfatning
1)
Støj på ion porte reducerer burst perioden.
2)
Blokering af bølgeudbredelse ved rumlig variation af den ATP
regulerende Na ion kanal.
3)
Koblingen mellem beta celler fører til en forøget excitation af
ellers inaktive celler.
Ref.: M.G. Pedersen and M.P. Sørensen, SIAM J. Appl. Math., 67(2), pp.530-542, (2007).
M.G. Pedersen and M.P. Sørensen, To appear in Jour. of Bio. Phys. Special issue on Complexity in Neurology and
Psychiatry, (2008).
1)
Bio-kemiske processer er meget komplekse og kræver omfattende
modellering.
2)
Simple og overskuelige modeller kan give kvalitativ indsigt.
3)
Der er lang vej til pålidelige kvantitative modeller.
4)
Matematiske modeller forventes dog at kunne bidrage til hurtigere
og mere sikker udvikling af medicin med færre dyreforsøg.
Studieretningsprojekter for gymnasiet
Se:
http://www.dtu.dk/Moed_DTU/Studieretningsprojekter.aspx