1 / 26 Systems Biology in Photosynthesis INTRODUCTION Rainer Machné Institute for Theoretical Chemistry, University of Vienna, Austria PSI - Photon System Instruments, Czech Republic Brno, April, 2011
2 / 26 Got headaches? normal people good in math it s called chlorophyll, not x... x=x x mathematicians chlorophyll=x... chlorophyll
/ 26 First-order kinetics: radioactive particles 1. Tōhoku earthquake & Tsunami released N radioactive particles from Fukushima power plant 2. The decay of an individual unstable nucleus is entirely random, a stochastic process : there is a certain probability that it happens in the next time-step t!. Decay rate is proportional to N and decay constant λ - the probability above - in [s 1 ]: dn(t) = λ N(t) Differential Equation N(t) 1 t N 0 N(t) dn(t) = λ remember high-school? t=0 N(t) =N 0 e λ t the SOLUTION!
4 / 26 First-order kinetics: radioactive particles 1. Tōhoku earthquake & Tsunami released N radioactive particles from Fukushima power plant 2. The decay of an individual unstable nucleus is entirely random, a stochastic process : there is a certain probability that it happens in the next time-step t!. Decay rate is proportional to N and decay constant λ - the probability above - in [s 1 ]: dn(t) = λ N(t) Differential Equation N(t) 1 t N 0 N(t) dn(t) = λ remember high-school? t=0 N(t) =N 0 e λ t the SOLUTION!
5 / 26 First-order kinetics: radioactive particles 1. Tōhoku earthquake & Tsunami released N radioactive particles from Fukushima power plant 2. The decay of an individual unstable nucleus is entirely random, a stochastic process : there is a certain probability that it happens in the next time-step t!. Decay rate is proportional to N and decay constant λ - the probability above - in [s 1 ]: dn(t) = λ N(t) Differential Equation N(t) 1 t N 0 N(t) dn(t) = λ remember high-school? t=0 N(t) =N 0 e λ t the SOLUTION!
6 / 26 First-order kinetics: radioactive particles 1. Tōhoku earthquake & Tsunami released N radioactive particles from Fukushima power plant 2. The decay of an individual unstable nucleus is entirely random, a stochastic process : there is a certain probability that it happens in the next time-step t!. Decay rate is proportional to N and decay constant λ - the probability above - in [s 1 ]: dn(t) = λ N(t) Differential Equation N(t) 1 t N 0 N(t) dn(t) = λ remember high-school? t=0 N(t) =N 0 e λ t the SOLUTION!
7 / 26 exponential decay N 0 400 800 0 20 40 60 80 100 time Analytic Integration: N(t) = N 0 e λ t Numeric Integration: N(t + t) = N(t) + dn t exemplified in MS Excel Spreadsheet: Bublina_CO2_generalpH_v7.xlsx
8 / 26 exponential decay N 0 400 800 t N N(0) = 1000 r = 0.1 0 20 40 60 80 100 time Analytic Integration: N(t) = N 0 e λ t Numeric Integration: N(t + t) = N(t) + dn t exemplified in MS Excel Spreadsheet: Bublina_CO2_generalpH_v7.xlsx
9 / 26 Growth 1 Bacterium 2 Bacteria: AUTO-CATALYSIS! Starting from N 0 Bacteria, with growth rate r, e. g. in h 1 dn(t) =r N(t) N(t) =N 0 e r t Limited by ressources (Verhulst 188): dn(t) =r N(t) (1 N(t) K c ) K c N(t) =N 0 e rt K c + N 0 (e rt 1) where K c is the carrying capacity of the environment r-strategy vs. K-strategy
10 / 26 exponential and limited growth N 0 40 80 Verhulst 188: 0 20 40 60 80 100 time N(0) = 1 r = 0.1 K = 100 K c N(t) =N 0 e rt K c + N 0 (e rt 1) Lotka (1910) & Volterra (1926): auto-catalytic reactions and predator-prey models: foxes eat rabbits less rabbits less foxes more rabbits: oscillatory feedback system - Hopf bifurcation!
11 / 26 exponential and limited growth N 0 40 80 Verhulst 188: 0 20 40 60 80 100 time N(0) = 1 r = 0.1 K = 100 K c N(t) =N 0 e rt K c + N 0 (e rt 1) Lotka (1910) & Volterra (1926): auto-catalytic reactions and predator-prey models: foxes eat rabbits less rabbits less foxes more rabbits: oscillatory feedback system - Hopf bifurcation!
12 / 26 The Chemostat GROWTH: substrate + cell 2 cells d[c] =µ [C] D [C] where µ is the growth rate in h 1, D is the dilution rate in h 1, [C] is biomass, e. g. in cells/l, grams of dry cell weight/l or in Carbon-mole C mol/l steady-state: dc = 0 µ = D Study growth in constant conditions! µ =µ 0 [S] [S] + K S where S is the substrate concentration in mol/l, and K S is a saturation constant, cf. Langmuir isotherms, Michaelis- Menten, etc. Cells as big auto-catalytic Enzymes! Monod J. 1942: The Growth of Bacterial Cultures. Ann. Rev. of Microbiology Eigen M. and Schuster PK. 1977: The hypercycle. A principle of natural self-organization. Naturwissenschaften
The (Photosynthetic) Growth of Bacterial Cultures 1 / 26
The (Photosynthetic) Growth of Bacterial Cultures 14 / 26
The (Photosynthetic) Growth of Bacterial Cultures 15 / 26
The (Photosynthetic) Growth of Bacterial Cultures 16 / 26
17 / 26 Back to Chemistry: Second-order Kinetics CO 2 + OH HCO Two stochastic processes: 1. Two randomly (Brownian) diffusing molecules collide: r [CO 2 ] [OH ] now: concentrations! 2. Colliding molecules react: r k Probability (stochastic view) or rate (deterministic view): r = k [CO 2 ] [OH ] in mol L 1 s 1
18 / 26 Back to Chemistry: Second-order Kinetics CO 2 + OH HCO Two stochastic processes: 1. Two randomly (Brownian) diffusing molecules collide: r [CO 2 ] [OH ] now: concentrations! 2. Colliding molecules react: r k Probability (stochastic view) or rate (deterministic view): r = k [CO 2 ] [OH ] in mol L 1 s 1
19 / 26 Back to Chemistry: Second-order Kinetics Bio-lingo: protein AsdF recruits protein HjkL to the membrane, transcription factor MIC activates gene umic Mesoscopic View: two stochastic processes: randomly diffusing molecules (Brownian movement) collide (1), and may react (2) Microscopic View: Arrhenius equation Ae Ea/RT, quantum chemistry Macroscopic View: deterministic description of average, for many molecules we can again use differential equations STOCHASTIC PROCESSES & SEQUENCE-BASED SPECIFICITY
20 / 26 CAVEATS CO 2 + OH HCO 1. In ionic solutions - such as cells or seawater - molecules are in interaction with the ions of the solutions r γ CO2 [CO 2 ] γ OH [OH ] 2. The laws of mass action kinetics are only valid in well-mixed D environments - but not e. g. on membranes or in the crowded cytoplams r [CO 2 ] h [OH ] h SUBJECT OF ONGOING RESEARCH! Elizalde MP. and Aparicio JL. Talanta 1995: Current theories in the calculation of activity coefficients-ii. [...] Schnell S. and Turner TE. Prog Biophys Mol Biol 2004: Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws.
21 / 26 Reaction Networks r j acidic medium alkaline medium rate rules, mol s 1 1 CO 2 + H 2 O HCO + H+ r 1 = k + 1 [CO 2] k [H + ] [HCO 1 ] CO 2 + OH HCO r = k + [OH ] [CO 2 ] k [HCO ] 2 HCO CO 2 + H + r 2 = k + 2 [HCO ] k [H + ] [CO 2 ] 2 4 HCO + OH CO 2 + H 2 O r 4 = k + 4 [OH ] [HCO ] k [CO 2 ] 4 w H 2 O H + + OH r w = k w + k w [H+ ] [OH ] d[co 2 ] = r 1 r d[hco ] = r 1 + r r 2 r 4 d[co 2 ] = r 2 + r 4 d[oh ] = r w r r 4 d[h + ] = r w + r 1 + r 2
22 / 26 Reaction Networks d[co 2 ] = (k + 1 [CO 2] k 1 [H+ ][HCO ]) (k+ [OH ][CO 2 ] k [HCO ]) d[hco ] = + (k + 1 [CO 2] k 1 [H+ ][HCO ]) + (k+ ([OH ][CO 2 ] k [HCO ]) d[co 2 ] d[oh ] (k + 2 [HCO ] k 2 [H+ ][CO 2 = + (k + 2 [HCO ] k 2 [H+ ][CO 2 ]) (k + 4 [OH ][HCO ] k 4 [CO2 ]) ]) + (k + 4 [OH ][HCO ] k 4 [CO2 ]) = + (k + w k w [H+ ][OH ]) (k + [OH ][CO 2 ] k [HCO ]) (k+ 4 [OH ][HCO ] k 4 [CO2 ]) d[h + ] = + (k + w k w [H+ ][OH ]) + (k + 1 [CO 2] k 1 [H+ ][HCO ]) + (k+ [OH ][CO 2 ] k [HCO ])
2 / 26 Reaction Networks: the Stoichiometry Matrix Stoichiometry matrix S: Reaction rates r: S r 1 r r 2 r 4 r w CO 2-1 -1 HCO +1 +1-1 -1 CO 2 +1 +1 OH -1-1 +1 H + +1 +1 +1 r 1 = k + 1 [CO 2] k 1 r = k + [OH ] [CO 2 ] k r 2 = k + 2 [HCO ] k 2 [H + ] [HCO ] r 4 = k + 4 [OH ] [HCO ] k 4 r w = k + w k w [H+ ] [OH ] [HCO ] [H + ] [CO 2 ] [CO 2 ] Generate ODE system: d C d t = S r
24 / 26 Reaction Networks: Fast Reactions in Equilibrium 1. Keep ph constant! 2. Assume equilibrium: K 2 = [CO2 ][H + ] [HCO ] and skip the reactions from and to CO 2!. Calculate lumped sum species: [HCO, ] = ([HCO ] + [CO2 ]) 4. In reaction rates, use: [HCO ] = [HCO, [H ] + ] [H + ] + K2 [OH ] = K w [H + ] [HCO ] = ([HCO ] + [H + ] [CO2 ]) [H + ] + K 2 (2) d[co 2 ] = (k + 1 d t [OH ])[CO 2 ] + (k + k 1 [H+ ])[HCO ] () d([hco ] + [CO2 ]) = + (k + 1 d t + k+ [OH ])[CO 2 ] (k + k 1 [H+ ])[HCO ] (4) (1)
25 / 26 Multiple Compartments CO bubble 2 CO liquid 2 CO liquid 2 CO bubble 2 e. g. mass transfer between gas and liquid k L a (k cc H r g = [CObubble 2 ] [CO liquid 2 ])
26 / 26 Multiple Compartments CO bubble 2 CO liquid 2 CO liquid 2 CO bubble 2 e. g. mass transfer between gas and liquid k L a (k cc H r g = [CObubble 2 ] [CO liquid 2 ]) 1. Formulate rate equations in substance time ( mol s ) instead of concentration time 2. Add division by the volumes: ( mol L s ) d C d t = S r v