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Its Upregulated in valve-forming LEC in vivo but {is not|isn't option is definitely the very simple exponential approach function A(t) A(1{e{ A ),Jtand geometrically corresponds to a family of curves parameterized by A and J (Figure 1b). With some measurements of Aalong the curve, parameters A and J can be estimated in a standard way: a least-squares fitting algorithm gives the best fit, and sensitivity Ses revealed very low levels of analysis or Monte Carlo simulations give the uncertainties. However, it helps to understand why KFP should work in this case. First, it is easy to see from Eq. 2 that parameter A determines the saturation level of Aand J=A determines the rate at which the saturation level is approached; in other words, A determines the scale and J=A determines the rate. To highlight this, we define a rate parameter, m:J=A; its inverse, tc :1=m A=J, is conventionally called the characteristic time-scaleFigure 1. Understanding KFP and rKFP. (a) A schematic diagram of KFP applied to a toy metabolic network. At t 0, the system is switched from a 12 C-labeled environment to 13 C-labeled one, and Ais measured at a few time points thereafter. (b) For a given trajectory of A(t) (the black solid curve), the three time regimes (linear, mixed and constant) are marked and three measurements are made (two in the mixed regime and one in the ^ constant). Normalizing it gives A(t) between 0 and 1 (the black dashed curve), parameterized by a single parameter m, which can be estimated by ^ comparing the normalized measurements to A's of different m's (the red and blue dashed curves). (c) A schematic diagram of rKFP applied to the same network in (a). Relative quantitation is performed on Ain two conditions (with subscripts x and y respectively) with the goal of estimating ax ay rJ Jy =Jx . (d) The ratio in m between ^(t) and ^(t) is rJ =rA (Eq. 6), and since m's and rA are identifiable from relative quantitation, so is rJ . doi:10.1371/journal.pcbi.1003958.gPLOS Computational Biology | www.ploscompbiol.orgRelative Changes of Metabolic Fluxesand numerically corresponds to the time needed to go from the 1 initial condition to 1{ ( 0:63) of the saturation level. Second, e like the familiar Michaelis-Menten hyperbolic curves, the exponential approach curves can also be thought as having three regimes, defined with respect to the characteristic time-scale: the linear regime when t tc , the constant regime when t tc , and the mixed regime in between when ttc (Figure 1b). Measurements of Ain the linear regime are informative of only J (for the slope of that regime is J), the constant regime of only A (for Ain that regime is constantly A), and the mixed regime of both. In practice, however, usually only the constant and mixed regimes are measured due to their experimental accessibility. Finally, after A is estimated from measurements in the constant regime, the estimation of J from measurements in the mixed regime can be understood in the following way: normalize all measurements by the estimated A to ^ describe the normalized variable A:A=A 1{e{mt ; parame^ terized only by m and now increasing from 0 to 1, Achanges from a sharply rising curve to a gently rising one as m decreases; the normalized measurements in the mixed regime nail down the ^ specific Awithin the family of curves, together with m an.T new biological insight.opposing depletion by the outflux, and their net difference describes the price of change of A.