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

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 μ 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 150mM Na K 110mM Na 30mM Na,K-pump K K K 3mM Na 10mM K-channel + Ca 2mM Ca-pump Ca Ca Ca 10 -4 mM

Red Blood Cell metabolism: Cell membrane + Na ATP Na + K K + Glycolysis + ADP Ca Ca Ca

Red Blood Cell metabolism: Inosine Hypoxantine Cell membrane + AMP Na ATP Na + degradation + K - K + AMP Glycolysis Adenylate kinase + AMP ADP Ca synthesis Ca Ca Adenosin Komarova S.V. et al, J.Theor. Biol. 1996, v.183, p.307-316 e Mosharov E.V. at al, FEBS Letters, 1998, v. 440, p.64-66 Adenine

Red Blood Cell metabolism:

Osmotic equations: � [ A ] � F �� � p e = exp � � � [ A ] R � � � p i [K + ] i +[Na + ] i � [A - ] i +ZW = 0 [K + ] i +[Na + ] i +[A - ] i + � +W = [K + ] e +[Na + ] e +[A - ] e = 2L = 300 mM P K = 1.24 � 10 -2 1/h; P Na = 1.22 � 10 -2 1/h; [K + ] e = 5 mM; [Na + ] e = 145 mM; [A - ] e = 150 mM

Osmotic equations: d V � � + = � � + [ Na ] 3 J ; � � i Na , K � ATPase Na 0 dt V � � � F � � F � � � � R + + J = P [ Na ] � [ Na ] exp � � � � Na Na e i � 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 ] � � PFK ALD 0 dt � V � � � d V � � [ DAP ] = � � � � � ALD TPI 0 dt V � � d � V � � � = � + � � � [ GAP ] � � ALD TPI GAPDH 0 dt V � � … … … … … … … …

Rates of enzymatic reactions (examples): 1 2 ([ G 6 P ] � [ F 6 P ] K ) / K GPI GPI � = � GPI GPI 2 3 1 + [ G 6 P ] / K + [ F 6 P ] / K GPI GPI 3 0 =360 mM/h, 1 2 =0.3 mM, =3, K =0.2 mM. � K K GPI GPI GPI GPI [ ] ( ) 3 3 + + + � 1 / 1 [ AMP ] / K 2 [ AMP ] /( K [ AMP ]) 1 . 1 [ ATP ][ F 6 P ] PFK PFK � = � ( ) ( ) PFK PFK 2 1 ( ) 4 � � K + [ ATP ] K + [ F 6 P ] 4 PFK PFK 1 + [ ATP ] / K � 8 PFK � 1 + 10 ( )( ) � � 4 4 3 5 1 + [ AMP ] / K 1 + [ F 6 P ] / K � � PFK PFK 0 1 2 K K � =380 mM/h, =0.1 mM, =2 mM, PFK PFK PFK 4 3 =19.5 � 10 -2 mM, K K =10 -2 mM, PFK 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

Na,K-pump 2,0 VOLUME V/V phys 1,5 Ca-activated K-channel 1,0 0,5 0,0 1 2 3 4 5 6 PERMEABILITY G/G phys Martinov M. et al. Biophys Chem, 1999, v.80, p.199-215

2,0 Cell death RBC volume V/V o 1,5 1,0 0,5 0,0 1 2 3 4 5 6 7 8 9 10 Permeability G/G phys

Red Blood Cell metabolism:

Steady-state fluxes should be equal 2u 1 = 2u 3 = u 7 = u 10 dATP/dt = 2u 1 - u consumption

U production U consumption u st Rate ATP st ATP

Glucose consumption rate (mmoles/ l cells*h) Rate (mM/ ��� ) 0,5 1,5 2,5 0 1 2 0 ATP concentration ( ��� ) [ATP] (mmoles/l c 500 1000 1500

Rate of glycolysis 200 200 2,5 2,5 Glucose consumption rate (%) Glucose consumption rate (%) Glucose consumption rate Glucose consumption rate 2 2 150 150 (mmoles/ l cells*h) (mmoles/ l cells*h) 1,5 1,5 100 100 1 1 50 50 0,5 0,5 0 0 0 0 0 50 100 150 0 50 100 150 0 500 1000 1500 0 500 1000 1500 ATP (%) [ATP] (mmoles/l ce ATP (%) [ATP] (mmoles/l ce 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 50 100 150 Energy charge ATP (%) Ataullakhanov F. et al. Eur J Biochem., 1981, v.115, p.359-365

+ + - - - Phosphofructo- - Hexokinase kinase Glucose Glucose G6P G6P F6P FDP F6P FDP ADP ATP ADP ATP ATP ADP ATP ADP Adenylate kinase AMP Adenosine AMP Adenosine IMP IMP NH 3 NH ADP ATP ADP ATP 3

Stable node Unstable node U st Rate Stable node ATP st ATP 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 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 ( � / � 0 ) • Similar decrease of HK 0.24-0.89 enzyme activity (5-20%) GPI 0.05-0.25 connected with hereditary PFK 0.08-0.60 anemia for almost all mutant ALD 0.04-0.16 enzyme TPI 0.016-0.30 GAPDH 0.20-0.50 • No correlation between PGK 0.01-0.30 decrease of activity and DPGP ─ severity of the anemia PGM ─ ENO 0.06-0.50 PK 0.05-0.40 LDH ─ Na,K-ATPase 0.20-0.60

Table 2. Comparison with experimental data Calculated Experimental activity data ( � cr / � 0 ) ( � / � 0 ) HK 0.39 0.24-0.89 GPI 0.015 0.05-0.25 PFK 0.011 0.08-0.60 ALD 0.03 0.04-0.16 TPI 0.0004 0.016-0.30 GAPDH 0.13 0.20-0.50 PGK 0.0033 0.01-0.30 DPGP 0.11 ─ PGM 0.0074 ─ ENO 0.20 0.06-0.50 PK 0.22 0.05-0.40 LDH 0.015 ─ Na,K-ATPase 0.11 0.20-0.60

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 Triosephosphateisomerase 83 120 550

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

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 = � o exp(-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 T T 1 o � 1 cr o cr m o = = � � = � � � = ( ) / ln( / ) dt exp( t / ) d t � � � � � � T T 0 0

Triosephosphateisomerase activity 100 Enzyme activity (%) Average enzyme activity 10 1 0,1 Critical Enzyme activity 0,01 (cell death) 0 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 Unstable Experimental enzyme enzyme data ( � cr / � 0 ) ( � m / � 0 ) ( � / � 0 ) HK 0.39 0.65 0.24-0.89 GPI 0.015 0.23 0.05-0.25 PFK 0.011 0.22 0.08-0.60 ALD 0.03 0.28 0.04-0.16 TPI 0.0004 0.13 0.016-0.30 GAPDH 0.13 0.43 0.20-0.50 PGK 0.0033 0.17 0.01-0.30 DPGP 0.11 0.40 ─ PGM 0.0074 0.20 ─ ENO 0.20 0.50 0.06-0.50 PK 0.22 0.52 0.05-0.40 LDH 0.015 0.23 ─ Na,K-ATPase 0.11 0.40 0.20-0.60