1. Synthesizing Units for modelling cell physiology Bas Kooijman Dept of Theoretical Biology Vrije Universiteit, Amsterdam http://www.bio.vu.nl/thb/deb/ Leiden, 2004/06/24 adult embryo juvenile Research program: Dynamic Energy Budget theory

2. Weird world at small scale Almost all transformations in cells are enzyme mediated Classic enzyme kinetics: based on chemical kinetics (industrial enzymes) diffusion/convection larger number of molecules constant reactor volume law of mass action: transformation rate  product of conc. of substrates Problematic application in cellular metabolism: definition of concentration (compartments, moving organelles) transport mechanisms (proteins with address labels, targetting, allocation) crowding (presence of many macro-molecules that do not partake in transformation) intrinsic stochasticity due to small numbers of molecules liquid crystalline properties surface area - volume relationships: membrane-cytoplasm; polymer-liquid connectivity (many metabolites are energy substrate & building block; dilution by growth) Alternative approach: reconstruction of transformation kinetics on the basis of cellular input/output kinetics

3. Self- ionization of water in cells A cell of volume 0.25  m 3 and pH 7 at 25°C has n = 15 protons N = 8 10 9 water molecules confidence intervals of pH 95, 90, 80, 60 % pH cell volume,  m 3 modified Bessel function 7

4. Diffusion cannot occur in cells

5. Crowding affects transport cytoskeletal polymers ribosomes nucleic acids proteins

6. ATP generation & use 5 10 6 ATP molecules in bacterial cell enough for 2 s of biosynthetic work Only used if energy generating & energy demanding transformations are at different site/time If ADP/ATP ratio varies, then rates of generation & use varies, but not necessarily the rates of transformations they drive Processes that are not much faster than cell cycle, should be linked to large slow pools of metabolites, not to small fast pools DEB theory uses reserve as large slow pool for driving metabolism

7. Yield vs growth 1/spec growth rate, 1/h 1/yield, mmol glucose/ mg cells Streptococcus bovis, Russell & Baldwin (1979) Marr-Pirt (no reserve) DEB spec growth rate yield Russell & Cook (1995): this is evidence for down-regulation of maintenance at low growth rates DEB theory: high reserve density gives high growth rates structure requires maintenance, reserves not

8. Methanotrophy ACAC Assim (catabolic)12-2000 A Assim (anabolic)010 MMaintenance010 GCGC Growth (catabolic)010 GAGA Growth (anabolic)001 CCarbon1100011 HHydrogen40203 OOxygen02120 NNitrogen00001 symbolprocessX: methane C: carbon dioxide H: waterO: dioxygenN: ammoniaE: reserve V: structure reserve density m E = M E /M V rate Yield coefficients T Chemical indices Macroscopic transformation (variable yield coefficients and indices): Microscopic transformations (constant coefficients and indices):

9. Methanotrophy spec growth rate, h -1 X/O N/O C/O flux ratio, mol.mol -1 spec flux, mol.mol -1.h -1 C E N X O X: methane C: carbon dioxide O: dioxygen N: ammonia E: reserve j EAm = 1.2 mol.mol -1.h -1 y EX = 0.8 y VE = 0.8 k M = 0.01 h -1 k E = 2 h -1 n HE = 1.8 n OE = 0.3 n NE = 0.3 n HV = 1.8 n OV = 0.3 n NV = 0.3 chemical indices Kooijman et al, 2004 Ecology, 85, 1230-1243

10. Enzyme kinetics A+B  C : conc of compounds A,B,C : fractions of bounded enzymes constant Synthesizing Unit Rejection Unit : dissociation rates: association rates

11. Isoclines for rate A+B  C.2.4.6.8 Conc A Conc B Synthesizing Unit Rejection Unit almost single substr limitation at low conc’s

12. Synthesizing units Generalized enzymes that process generalized substrates and follow classic enzyme kinetics E + S  ES  EP  E + P with two modifications: back flux is negligibly small E + S  ES  EP  E + P specification of transformation is on the basis of arrival fluxes of substrates rather than concentrations In spatially homogeneous environments: arrival fluxes  concentrations

13. Simultaneous Substrate Processing Chemical reaction: 1A + 1B 1C Poisson arrival events for molecules A and B blocked time intervals acceptation event ¤ rejection event Flux of C: production Kooijman, 1998 Biophys Chem 73: 179-188

14. SU kinetics: n 1 X 1 +n 2 X 2  X 0tbtb tctc time product release product release binding prod. cycle flux of X max flux of X Expected value of t b Survivor function of t b Period between subsequent arrivals is exponentially distributed Sum of exponentially distributed vars is gamma distributed Production flux not very sensitive for details of stoichiometry Stoichiometry mainly affects arrival rates Kooijman, 1998 Biophys Chem 73: 179-188

15. Simultaneous Nutrient Limitation Specific growth rate of Pavlova lutheri as function of intracellular phosphorus and vitamin B 12 at 20 ºC Data from Droop 1974 Note the absence of high contents for both compounds due to damming up of reserves, and low contents in structure (at zero growth) P content, fmol/cell B 12 content, 10 -21 mol/cell Kooijman, 1998 Biophys Chem 73: 179-188

16. Reserve interactions 5.2.4 Spec growth rate, d -1 P-content, fmol.cell -1 P-conc, μM B 12 -conc, pM B 12 -cont., 10 -21.mol.cell -1 PVitamin B 12 kEkE 1.191.22 d -1 y XV 0.39 10 - 15 2.35 mol.cell -1 j EAm 4.91 10 - 21 76.6 10 -15 mol.cell -1. d -1 κEκE 0.690.96 kMkM 0.00790.135 d -1 K0.0170.12 pM, μM Data from Droop 1974 on Pavlova lutheri P(μM)B 12 (pM) 1.4468 14.46.8 1.4420.4 1.446.8

17. C,N,P-limitation Nannochloropsis gaditana (Eugstimatophyta) in sea water Data from Carmen Garrido Perez Reductions by factor 1/3 starting from 24.7 mM NO 3, 1.99 mM PO 4 N,P reductions N reductions P reductions

18. C,N,P-limitation Nannochloropsis gaditana in sea water For DIC nitrate phosphate res. dens. structure uptake rate spec growth rate spec growth

19. Producer/consumer dynamics producer consumer nutr reserve of producer : total nutrient in closed system : hazard rate special case: consumer is not nutrient limited spec growth of consumer Kooijman et al 2004 Ecology, 85, 1230-1243

20. Producer/consumer dynamics Consumer nutrient limited Consumer not nutrient limited Hopf bifurcation Hopf bifurcation tangent bifurcation transcritical bifurcation homoclinic bifurcation

21. Interactions of substrates Kooijman, 2001 Phil Trans R Soc B 356: 331-349

22. Photosynthesis 2 H 2 O + 4 h  O 2 + 4 H + + 4 e - CO 2 + 4 H + + 4 e -  CH 2 O + H 2 O CO 2 + H 2 O + light  CH 2 O + O 2

23. no synthesis of hydrocarbons at compensation point Photorespiration RuP 2 ribulose 1,5-biphosphate (C 5 + C  2 C 3 ) (C 5  C 3 + C 2 ) Transformations are catalized by Rubisco, which evolved in anaerobic environments O 2 competes with CO 2 which gives an oxidation, rather than a reduction

24. Co-metabolism Consider coupled transformations A  C and B  D Binding probability of B to free SU differs from that to SU-A complex

25. Co-metabolism Co-metabolic degradation of 3-chloroaniline by Rhodococcus with glucose as primary substrate Data from Schukat et al, 1983 Brandt et al, 2003 Water Research 37, 4843-4854

26. Co-metabolism Co-metabolic anearobic degradation of citrate by E. coli with glucose as primary substrate Data from Lütgens and Gottschalk, 1980 Brandt et al, 2003 Water Research 37, 4843-4854

27. Adaptation glucose, mg/l specific growth rate, h -1 “wild type” Schulze & Lipe, 1964 glucose-adapted Senn, 1989 Glucose-limited growth of Escherichia coli 70 mg/l 0.06 mg/l max.5 max many types of carriers only carriers for glucose

28. Inhibition A does not affect B in y AC A  C; B inhibits binding of A unbounded fraction binding prob of A arrival rate of A dissociation rate of A yield of C on A A inhibits binding of B in y AC A  C; B inhibits binding of A

29. Adaptation Batch culture, Monod special case Model elements: uptake of substrate by specific carriers carrier densities n A and n B metabolic signals from uptake  f i n i relative signal s A = p A f A n A /  i p i f i n i carrier production by SUs that are fed by relative signals that inhibit reciprocally carriers have a common turnover rate Result: Expression fraction  0 asymptotically in absence of substrate biomass density substrate i conc scaled func response saturation coeff for i yield of biom on substr spec growth rate max spec growth rate on i expression fraction for i carrier turnover rate preference ratio Brandt et al, 2004 Water Research, 38, 1003 - 1013

30. Diauxic growth time, h biomass conc., OD 433 acetate oxalate Substrate conc., mM Growth of acetate-adapted Pseudomonas oxalaticus OX1 data from Dijkhuizen et al 1980 SU-based DEB curves fitted by Bernd Brandt Adaptation to different substrates is controlled by: enzyme turnover 0.15 h -1 preference ratio 0.5 cells Brandt et al, 2004 Water Research, 38, 1003 - 1013

31. Diauxic growth biomass conc., OD 590 Growth of succinate-adapted Azospirillum brasilense intracellular amounts followed with radio labels data from Mukherjee & Ghosh 1987 Adaptation to different substrates is controlled by: enzyme turnover 0.7 h -1 preference ratio 0.8 time, h fructose conc, mM succinate conc, mM succinate fructose cells suc in cells fruc in cells Brandt et al, 2004 Water Research, 38, 1003 - 1013

32. Social inhibition of x  e sequential parallel dilution rate substrate conc. biomass conc. No socialization Implications: stable co-existence of competing species “survival of the fittest”? absence of paradox of enrichment x substrate e reserve y species 1 z species 2 Collaboration: Van Voorn, Gross, Feudel, Kooi, Kooijman

33. Aggressive competition V structure; E reserve; M maintenance substrate priority E  M; posteriority V  M J E flux mobilized from reserve specified by DEB theory J V flux mobilized from structure  amount of structure (part of maint.) excess returns to structure k V dissociation rate SU-V complex k E dissociation rate SU-E complex k V k E depend on  such that k M = y ME k E (  E. +  EV )+y MV k V .V is constant J E M, J V M JEJE k V = k E k V < k E Collaboration: Tolla, Poggiale, Auger, Kooi, Kooijman