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Mathematical models of erythrocyte. What they give us for understanding the disorders and ageing of this cell
Fazly Ataullakhanov

Center for Theoretical Problems of Physico-Chemical Pharmacology, National Research Center for Hematology, Lomonosov Moscow State University Moscow, October 2008


In memory of Anatol Zhabotinsky


Questions: · What is a disease from the mathematical point of view? · Complex and simple models: how do they relate with each other? ·Why do so many cellular enzymes have excessively high activities?


Topics of this lecture: · Red blood cell (RBC) ­ an overview · Red blood cell ­ metabolism and viability; ageing · A mathematical model · Hereditary anemia due to enzyme deficiency: key and non-key enzymes · Modeling of viability of the red blood cells with unstable mutant forms of enzymes



Red Blood Cell: · Flexible flat cell about 8 in diameter, · No nucleus, · No protein syntesis · Hemoglobin content > 98% · Metabolic networks contain about 200 enzymes


Red Blood Cell: Hemoglobin content > 98% Redox control Osmotic control -> volume stabilization


Red Blood Cell metabolism: Cell membrane

Na
Na,K-pump
K

Na

K 110mM

Na 150mM 30mM K 3mM Ca 2mM

K
+ Ca

K-channel Ca-pump

Na 10mM

Ca

Ca 10-4mM


Red Blood Cell metabolism: Cell membrane
+

Na
K

Na +

ATP
Glycolysis

K
+

+

Ca

Ca

ADP

Ca


Red Blood Cell metabolism: Cell membrane
+

Inosine Hypoxantine

Na
K

Na +

ATP
Glycolysis

+

AMP degradation -

K
+

+

AMP
Adenylate kinase

Ca

Ca

ADP

Ca

AMP synthesis

Komarova S.V. et al, J.Theor. Biol. 1996, v.183, p.307-316 Mosharov E.V. at al, FEBS Letters, 1998, v. 440, p.64-66

Adenosin e Adenine


Red Blood Cell metabolism:


Osmotic equations:

[ Ap ]e [ Ap ]i

= exp

F R

[K+]i+[Na+]i [A-]i+ZW = 0 [K+]i+[Na+]i+[A-]i+ +W = [K+]e+[Na+]e+[A-] = 2L = 300 mM
PK=1.24 10-2 1/h; PNa=1.22 10-2 1/h; [K+]e= 5 mM; [Na+]e=145 mM; [A-]e=150 mM
e


Osmotic equations:

d V + [ Na ]i 0 = 3 Na ,K ATPase + J Na ; dt V F F + + R [ Na ]e [ Na ]i exp J Na = PNa F R exp 1 R

d V + [ K ]i 0 = ... dt V d V ++ [Ca ]i 0 = ... dt V


Metabolic equations (examples):

d V [FDP] = dt V0 d V [DAP] = dt V0

PFK

ALD

ALD

TPI

d V [GAP] 0 = dt V

ALD

+

TPI

GAPDH

........................


Rates of enzymatic reactions (examples):
([G 6P] [F6P]K 1 ) / GPI = GPI GPI 2 1 + [G 6P] / K GPI + [F6P]
0 =360 mM/h, GPI

2 K GPI

/ K3 GPI
2 K GPI =0.3 mM,

K1 =3, GPI

K 3 =0.2 mM. GPI

PFK = PFK

1.1 [ATP][F6P]

( K

2 PFK

+ [ATP] K 1 PFK + [ F6P

) (

[/( 11 ])

3 + [AMP] / K 3 PFK + 2[ AMP] /( K PFK + [ AMP])

)

]

1 + 10 8

( 1

( 1

+ [ATP] / K 4 PFK

+ [AMP] / K 3 PFK

)( 1

4

)

4

+ [F6P] / K 5 PFK

)

4

0 PFK =380 mM/h,

K1 PFK =0.1 mM,
-2 K4 PFK =19.5 10 mM,

K 2 =2 mM, PFK

K 3 =10-2mM, PFK

........................


Anemia ­ low RBC content in the blood Hemopoiesis Cell death

RBC

Red blood cell death caused mostly by osmotic swelling Osmotic swelling caused by decrease of the enzyme activity Hereditary anemia due to enzyme deficiency ­> caused by increased rate of a cell death


2,0
phys

Na,K-pump

VOLUME V/V

1,5 1,0 0,5 0,0 1 2 3

Ca-activated K-channel

4

5
phys

6

PERMEABILITY G/G

Martinov M. et al. Biophys Chem, 1999, v.80, p.199-215


2,0

Cell death

RBC volume V/Vo

1,5 1,0 0,5 0,0 1 2 3 4 5 6 7 8
phys

9

10

Permeability G/G


Red Blood Cell metabolism:


Steady-state fluxes should be equal 2u1 = 2u3 = u7 = u10

dATP/dt = 2u1- u

consumption


U

production

Rate

u

U
st

consumption

ATPst ATP


2,5

Glucose consumption rate (mmoles/ Rate (mM/l cells*h) )

2

1,5

1

0,5

0 0 500 1000 1500 [ATP] (mmoles/l ( ATP concentration c

)


Rate of glycolysis
2,5
Glucose consumption rate (%) Glucose consumption rate (%)

200

Glucose consumption rate Glucose consumption rate (mmoles/ cells*h) (mmoles/ ll cells*h)

2

150

1,5

100

1

0,5

50

0 0 500 1000 1500 [ATP] (mmoles/l ce

0 0 50 100 ATP (%) 150

ATP concentration

energy charge

Ataullakhanov F. et al. Eur J Biochem., 1981, v.115, p.359-365


Energy charge is one of the few essential variables:

=

ATP + 0.5 ADP ATP + ADP + AMP


200

Glucose consumption rate (%) Rate of glycolysis

150

100

50

0 0 50

Energy charge ATP (%)

100

150

Ataullakhanov F. et al. Eur J Biochem., 1981, v.115, p.359-365


Hexokinase

+

F6P

Glucose
ATP

G6P
ADP

Phosphofructokinase

FDP
Adenylate kinase

ATP

ADP

Adenosine
ATP

AMP
ADP

IMP NH3


Stable node Unstable node

Rate

U

st

Stable node

ATPst
Unstable node Stable node

ATP


Hexokinase


Panel a: (1) G6P, (2) 2,3-DPG, (3) ATP.

Panel b: (1) intracellular Na, (2) erythrocyte volume, (3) the total concentration of osmotically active metabolites

Martinov M. et al. BBA, 2000, v.1474, p.75-87



[ATP]/ [ATPo]


Table 1. Decrease in enzyme activity in the blood of patients with hereditary anemia
( HK GPI PFK ALD TPI GAPDH PGK DPGP PGM ENO PK LDH Na,K-ATPase

/

0)

0.24-0.89 0.05-0.25 0.08-0.60 0.04-0.16 0.016-0.30 0.20-0.50 0.01-0.30 0.06-0.50 0.05-0.40 0.20-0.60

· Similar decrease of enzyme activity (5-20%) connected with hereditary anemia for almost all mutant enzyme · No correlation between decrease of activity and severity of the anemia


Table 2. Comparison with experimental data
Calculated activity ( HK GPI PFK ALD TPI GAPDH PGK DPGP PGM ENO PK LDH
cr/ 0

Experimental data (/
0

)

)

0.39 0.015 0.011 0.03 0.0004 0.13 0.0033 0.11 0.0074 0.20 0.22 0.015

0.24-0.89 0.05-0.25 0.08-0.60 0.04-0.16 0.016-0.30 0.20-0.50 0.01-0.30 0.06-0.50 0.05-0.40 0.20-0.60

Na,K-ATPase 0.11


12 360 380 76 3000 690 7330 1100 83 120 550


Dibrov B. et al., have shown that the range of dynamic stability can be widened greatly, if the pathway contains one or two reactions (but not more) with relatively small effective rate constants.

Dibrov B. et al., J. Math. Biology (1982) v.15, p.51-63


12 360 380 76 3000 690 7330 1100 83 120 550

Triosephosphateisomerase


Triosephosphateisomerase
80

[DAP]/[DAP]

60

o

40

20

0 1E-4

1E-3

0,01
TPI/ o TPI

0,1

1


Hypothesis:

Mutant form of an enzyme is unstable and decays exponentially:

(t) =

exp(-t/ )

Erythrocyte dies when activity of the mutant enzyme decreases down to cr = oexp(-T/t),
where T is an RBC's lifespan in circulation

So T = t ln( o/ cr) , and the mean value of enzyme activity in the blood is
m

=1 T

T

0

1 dt = T

T o 0

exp( t / )dt = ( =

o

cr

) / ln(

o

/

cr

)


Triosephosphateisomerase activity

100
Enzyme activity (%)

10 1 0,1 0,01 0

Average enzyme activity

Critical Enzyme activity (cell death)

10 20 30 40 Age of Red Blood Cells (days)


Hypothesis:

Mutant form of an enzyme is unstable and decays exponentially.
Predictions:

· Mean level of enzyme activity in the blood is much higher than critical and falls in a diapason of 5-20%. · Severity of anemia should correlate with the rate of enzyme degradation in the cell but not with the mean enzyme activity.


Table 2. Comparison with experimental data
Stable enzyme ( HK GPI PFK ALD TPI GAPDH PGK DPGP PGM ENO PK LDH
cr/ 0

Unstable enzyme (
m

Experimental data (/
0

)

/

0

)

)

0.39 0.015 0.011 0.03 0.0004 0.13 0.0033 0.11 0.0074 0.20 0.22 0.015

0.65 0.23 0.22 0.28 0.13 0.43 0.17 0.40 0.20 0.50 0.52 0.23 0.40

0.24-0.89 0.05-0.25 0.08-0.60 0.04-0.16 0.016-0.30 0.20-0.50 0.01-0.30 0.06-0.50 0.05-0.40 0.20-0.60

Na,K-ATPase 0.11


Conclusions: · What is a disease from the mathematical point of view? Existence in the vicinity of a bifurcation. · Complex and simple models: how do they relate with each other? Simple model helps to understand a nature of bifurcation, thereby helping to interpret a more complete, complex quantitative model. · Why do so many cellular enzymes have excessively high activities? To allow a high degree of stabilization control.


Contributors:

M. V. B. A.

Martinov, Vitvitsky, Dibrov Zhabotinsky