Mathematical models of erythrocyte. What they give us for - - PowerPoint PPT Presentation

Fazly Ataullakhanov

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

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

In memory of Anatol Zhabotinsky

• What is a disease from the mathematical point of

view? Questions:

• Complex and simple models: how do they relate

with each other?

• Why do so many cellular enzymes have excessively

high activities?

• 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 Topics of this lecture:

Red Blood Cell:

• Flexible flat cell about 8 μ in diameter,
• No nucleus,
• No protein syntesis
• Hemoglobin content > 98%
• Metabolic networks contain

about 200 enzymes

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

Ca

Ca K Na

K Na

+

Na 150mM K 3mM Ca 2mM 30mM K 110mM Na 10mM Ca 10-4mM Cell membrane Na,K-pump Ca-pump K-channel Red Blood Cell metabolism:

Ca Glycolysis

K Na

K Na

+

ATP ADP

+ +

Ca

Ca +

Cell membrane Red Blood Cell metabolism:

Ca Glycolysis

AMP

Adenylate kinase K Na

K Na

+

ATP ADP

+ +

Ca

Ca +

AMP degradation AMP synthesis

Adenosin e Adenine Inosine Hypoxantine

Cell membrane

+

• Red Blood Cell metabolism:

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

Red Blood Cell metabolism:

• =
• R

F A A

i p e p

exp ] [ ] [

[K+]i+[Na+]i[A-]i+ZW = 0

[K+]i+[Na+]i+[A-]i++W = [K+]e+[Na+]e+[A-]e = 2L = 300 mM

PK= 1.2410-2 1/h; PNa= 1.2210-2 1/h; [K+] e= 5 mM; [Na+] e= 145 mM; [A-] e= 150 mM Osmotic equations:

• =

+

• =
• +

+

• +
• R

F Na Na R F R F P J J V V Na dt d

i e Na Na Na ATPase K Na i

exp ] [ ] [ 1 exp ; 3 ] [

,

... ] [ =

• +

V V K dt d

i

... ] [ =

• +

+

V V Ca dt d

i

Osmotic equations:

ALD PFK

V V ] FDP [ dt d

• =
• TPI

ALD

V V ] DAP [ dt d

• =
• GAPDH

TPI ALD

V V ] GAP [ dt d

• +
• =
• …

… … … … … … …

Metabolic equations (examples):

… … … … … … … …

3 GPI 2 GPI 2 GPI 1 GPI GPI GPI

K / ] P 6 F [ K / ] P 6 G [ 1 K / ) K ] P 6 F [ ] P 6 G ([ + +

• =
• GPI
• 1

GPI

K

2 GPI

K

3 GPI

K

( ) ( ) ( )

[ ]

( ) ( )( )

• +

+ + + + + + + +

• =
• 4

5 PFK 4 3 PFK 4 4 PFK 8 3 PFK 3 PFK 1 PFK 2 PFK PFK PFK

K / ] P 6 F [ 1 K / ] AMP [ 1 K / ] ATP [ 1 10 1 ]) AMP [ K /( ] AMP [ 2 K / ] AMP [ 1 / 1 ] P 6 F [ K ] ATP [ K ] P 6 F ][ ATP [ 1 . 1

PFK

• 1

PFK

K

2 PFK

K

3 PFK

K

4 PFK

K

=360 mM/h, =3, =0.3 mM, =0.2 mM.

=380 mM/h, =0.1 mM, =2 mM, =10-2mM,

=19.510-2mM,

Rates of enzymatic reactions (examples):

Osmotic swelling caused by decrease of the enzyme activity Anemia – low RBC content in the blood

RBC

Hemopoiesis Cell death Red blood cell death caused mostly by osmotic swelling Hereditary anemia due to enzyme deficiency –> caused by increased rate of a cell death

1 2 3 4 5 6 0,0 0,5 1,0 1,5 2,0 PERMEABILITY G/G phys VOLUME V/V phys Na,K-pump Ca-activated K-channel

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

1 2 3 4 5 6 7 8 9 10 0,0 0,5 1,0 1,5 2,0

Permeability G/G phys RBC volume V/Vo Cell death

Red Blood Cell metabolism:

dATP/dt = 2u1- uconsumption Steady-state fluxes should be equal 2u1 = 2u3 = u7 = u10

ATPst U production ATP ust Uconsumption Rate

0,5 1 1,5 2 2,5 500 1000 1500 [ATP] (mmoles/l c Glucose consumption rate

(mmoles/ l cells*h)

ATP concentration () Rate (mM/)

ATP concentration energy charge Rate of glycolysis

50 100 150 200 50 100 150 ATP (%) Glucose consumption rate (%) 0,5 1 1,5 2 2,5 500 1000 1500 [ATP] (mmoles/l ce Glucose consumption rate

(mmoles/ l cells*h)

50 100 150 200 50 100 150 ATP (%) Glucose consumption rate (%) 0,5 1 1,5 2 2,5 500 1000 1500 [ATP] (mmoles/l ce Glucose consumption rate

(mmoles/ l cells*h)

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

• =

=

ATP + 0.5 ADP ATP + ADP + AMP Energy charge is one of the few essential variables:

50 100 150 200 50 100 150 ATP (%) Glucose consumption rate (%)

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

Energy charge Rate of glycolysis

F6P F6P

FDP FDP

ATP ATP ADP ADP AMP AMP

Adenosine Adenosine

IMP IMP

• +

+

Phosphofructo- kinase Adenylate kinase

NH NH3

3

ATP ATP ADP ADP

Glucose Glucose G6P G6P

Hexokinase

• ATP

ATP ADP ADP

ATPst Rate ATP Ust

Stable node Unstable node Stable node Unstable node Stable node

Hexokinase

Panel a: (1) G6P, (2) 2,3-DPG, (3) ATP. Panel b: (1) intracellular Na, (2) erythrocyte volume, (3) the total concentration

• f osmotically active

metabolites

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

[ATP]/ [ATPo]

0.20-0.60 Na,K-ATPase ─ LDH 0.05-0.40 PK 0.06-0.50 ENO ─ PGM ─ DPGP 0.01-0.30 PGK 0.20-0.50 GAPDH 0.016-0.30 TPI 0.04-0.16 ALD 0.08-0.60 PFK 0.05-0.25 GPI 0.24-0.89 HK (/0)

Table 1. Decrease in enzyme activity in the blood of patients with hereditary anemia

• 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

0.20-0.60 0.11 Na,K-ATPase ─ 0.015 LDH 0.05-0.40 0.22 PK 0.06-0.50 0.20 ENO ─ 0.0074 PGM ─ 0.11 DPGP 0.01-0.30 0.0033 PGK 0.20-0.50 0.13 GAPDH 0.016-0.30 0.0004 TPI 0.04-0.16 0.03 ALD 0.08-0.60 0.011 PFK 0.05-0.25 0.015 GPI 0.24-0.89 0.39 HK (/0) (cr/0) Experimental data Calculated activity

Table 2. Comparison with experimental data

12 360 380 76 3000 690 7330 1100 83 120 550

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

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.

12 360 380 76 3000 690 7330 1100 83 120 550

Triosephosphateisomerase

1E-4 1E-3 0,01 0,1 1 20 40 60 80 [DAP]/[DAP]

• TPI/
• TPI

Triosephosphateisomerase

cr = oexp(-T/t),

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

=

• =

=

• T
• T

m

d t T dt T t ) / exp( 1 1

• )

/ ln( / ) (

• cr
• cr

=

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

where T is an RBC’s lifespan in circulation

10 20 30 40 0,01 0,1 1 10 100

Age of Red Blood Cells (days) Enzyme activity (%)

Critical Enzyme activity (cell death) Average enzyme activity

Triosephosphateisomerase activity

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.

0.20-0.60 0.40 0.11 Na,K-ATPase ─ 0.23 0.015 LDH 0.05-0.40 0.52 0.22 PK 0.06-0.50 0.50 0.20 ENO ─ 0.20 0.0074 PGM ─ 0.40 0.11 DPGP 0.01-0.30 0.17 0.0033 PGK 0.20-0.50 0.43 0.13 GAPDH 0.016-0.30 0.13 0.0004 TPI 0.04-0.16 0.28 0.03 ALD 0.08-0.60 0.22 0.011 PFK 0.05-0.25 0.23 0.015 GPI 0.24-0.89 0.65 0.39 HK (/0) (m/0) (cr/0) Experimental data Unstable enzyme Stable enzyme

Table 2. Comparison with experimental data

• What is a disease from the mathematical point of

view? Conclusions: 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. Martinov,
• V. Vitvitsky,
• B. Dibrov
• A. Zhabotinsky