28 Net1: Today's story & goals • Macroscopic continuous concentration rates – Cooperativity & Hill coefficients – Bistability • Mesoscopic discrete molecular numbers – Approximate & exact stochastic • Chromosome Copy Number Control • Flux balance optimization – Universal stoichiometric matrix – Genomic sequence comparisons
29 Copy Number Control Models • Replication of ColE1 & R1 Plasmids • Determine the factors that govern the plasmid copy number – cellular growth rate – One way to address this question is via the use of a kinetic analysis of the replication process, and relate copy number to overall cellular growth. • Why? the copy number can be an important determinant of cloned protein production in recombinant microorganisms
30 RNA II RNA I RNA Polymerase Rom protein RNA II RNA I RNase H DNA Polymerase ColE1 CNC mechanism Rnase H cleaved RNAII forms a primer for DNA replication RNA I binding to RNA II prevents RNaseH from cleaving RNA II
31 Where do we start? Dynamic mass balance What are the important parameters? Plasmid, RNA I, RNA II, Rom, All the constants degradation, initiation, inhibition RNaseH rate is very fast instantaneous DNA polymerization is very rapid Simplify by subsuming [RNA II] model RNA I inhibition RNA I and RNA II transcription is independent (neglect convergent transcription) Rom protein effects constant Consider 2 species: RNA I and plasmid Many more assumptions... Assumptions? RNA II RNA I RNA Polymerase Rom RNA II RNA I RNase H DNA Polymerase
32 Dynamic Mass Balance: ColE1 RNAI [concentration in moles/liter] R k N k dt dR d ) ( 1 Rate of change of [RNA I] Synthesis of RNA I Degradation of RNA I Dilution due to cell growth = - - R = [RNA I] k 1 = rate of RNA I initiation N = [plasmid] k d = rate of degradation = growth rate Keasling,& Palsson (1989) J theor Biol 136, 487-492; 141, 447-61.
33 Dynamic Mass Balance: ColE1 Plasmid Rate of change of [N] Plasmid Replication Dilution due to cell growth = - R = [RNA I] k 2 = rate of RNA II initiation N = [plasmid] K I = RNA I/RNA II binding constant (an inhibition constant) = growth rate N N R K k dt dN I ) 1 1 ( 2 Solve for N(t).
34 Mathematica ODE program Formulae for steady state start at mu=1 shift to mu=.5 and then solve for plasmid concentration N as a function of time.
35 Stochastic models for CNC Paulsson & Ehrenberg, J Mol Biol 1998;279:73-88. Trade-off between segregational stability and metabolic burden: a mathematical model of plasmid ColE1 replication control. (Pub), J Mol Biol 2000;297:179-92. Molecular clocks reduce plasmid loss rates: the R1 case. (Pub) While copy number control for ColE1 efficiently corrects for fluctuations that have already occurred, R1 copy number control prevents their emergence in cells that by chance start their cycle with only one plasmid copy. Regular, clock- like, behaviour of single plasmid copies becomes hidden in experiments probing collective properties of a population of plasmid copies ... The model is formulated using master equations, taking a stochastic approach to regulation”
36 From RBC & CNC to models for whole cell replication?
- Fall '19
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