Life on planet earth may have started with certain "self-replicating" macromolecules (some ancient form of RNA perhaps?) which had the capability to catalyze their own synthesis from simpler "food molecules" (such as RNA bases, sugars, phosphates). Consider a reaction network model for a very simple self-replicator molecule S which produces a copy of itself by catalyzing the dimerization of two food molecules, F and G, to form a new S-molecule, in three reaction steps, r1, r2 and r3:
r1: S + F + G ó SFG
r2: SFG ó SS
r3: SS ó S + S
Each reaction step represents a reaction pair, comprising both a forward reaction (denoted by r1+, r2+, r3+) and a backward reaction (denoted by r1-, r2-, r3-).
For example,
in the r1+ reaction, S+F+G⇒SFG, an F and a G a bind to an S to form a intermediate "molecular complex", SFG, where F and G are attached to S, but F and G have not yet bound to each other to form a new S. In the r2+ reaction, SFG⇒SS, the F and the G, while being attached to the original S, bind to each other to form a new "child" S. The new child S is then still then attached to the original "parent" S, with parent and child forming another intermediate molecular complex, called SS. In the r3+ reaction, SS⇒S+S, the parent-child complex SS breaks apart, with the child S being released from the parent S. Note that child and parent are indistinguishable molecules; hence they are denoted by the same symbol S here. In the problem statements below, underlined text indicates mandatory elements you must include in your report to get a full credit.
(a) Explain briefly, in words, what happens in each of the three backward reactions, r1-, r2-, and r3- , analogous to the description of r1+, r2+, and r3+ given above.
(b) Draw a reaction network diagram for the network consisting of the foregoing three reaction pairs (r1, r2, r3), analogous to the H2-combustion network diagrams discussed in class and in your homework assignment. The diagram should consist of boxes, circles and arrows, with boxes representing molecular species; with circles representing reaction pairs; and with arrows, pointing from/to boxes and to/from circles, representing the flow of reactants/products into/out of reactions, for the case of the forward reaction of each reaction pair. State clearly how many boxes, circles and arrows you need to draw this diagram. Make sure all boxes and circles are clearly labeled by the molecular species name and the reaction pair name, respectively.
(c) Write down the system of kinetic rate equations for the net rates of production of all molecular species in the network: d[S]/dt=..., d[F]/dt=..., etc. Use the symbols k1+, k2+, k3+ to denote the rate coefficients (RCs) of the forward reactions r1+, r2+, r3+; and k1-, k2-, k3- for the RCs of the backward reactions r1-, r2-, r3-, respectively.
(d) Make Figure 1 showing plots of all species concentrations [....] vs. time t. Use the given Excel file that has the concentrations of all species(S, F, G, SFG, and SS) to make the plots.
(e) For the t-dependent [S]-data in the given Excel-file, use your spreadsheet to calculate eq.1: Y := ln([S]) at all listed times t, where "ln(...)" means "natural logarithm. Make a Figure 2 showing a plot of Y vs. t, with (t,Y) data points plotted as symbols(use scattered data plotter in the Excel file) only, without a line, with axes clearly labeled and a figure caption explaining what data is shown in the plot. A detailed mathematical theory for this network model predicts that [S] grows exponentially with time t for "early times", i.e., eq.2: [S] ≈ B egt as long as [S] with some constant pre-factor B and some constant "growth rate coefficient" g. If eq.2 is correct, what should be the shape of the Y-vs.-t curve in Figure 2 at early times? Are the numerical early-time results plotted in Figure 2 consistent with the theory prediction of eq.2 ? If so, explain why; if not, explain why not. Hint: Recall that ln(B C)=ln(B)+ln(C) and ln(ex)=x.
(f) Estimate the prefactor B and and growth rate coefficient g by fitting a straight line, Y=at+b, to the early-time Y-vs.-t data used in Figure 2. Do this by using your spreadsheet to perform a linear regression, a.k.a. a linear least squares fit, as illustrated in the "LeastSquaresFit" sample files posted on Elearning. Assuming eq.2 is correct, how are the slope a and ordinate intercept b of this straight-line fit related to g and B? Use these relations to estimate g and B from the slope and intercept of the straight-line-fit. In Figure 2, add the fitted straight line, Y=at+b, to the plot, shown as a line only, without symbols. State clearly, in the figure caption, which (t,Y)-data points from Figure 2 (i.e. from which t-range) were included in your linear regression analysis. (Note: you should only include data points that, by visual inspection, are reasonably close to falling on the straight line.) The detailed mathematical theory for this network model also predicts that eq.3: g ≈ k1+ [F]o [G]o and B ≈ [F]o [G]o. Where k1+=0.3 and [F]o, [G]o, and [S]o are the initial concentration at time t=0 Make a Table 1 to compare the numerical results for g and B (obtained from the straight-line fit) to the theoretical results predicted by eq.3. Do they agree? Make sure that all columns/rows are clearly labeled in Table 1 and explained in the table caption.