Higher Precision in Initial rates may be achievable: A test of a Pseudo-statistical method
BackgroundThere has been a concerted effort at establishing the best method for the measurement of initial rates for various purposes, including the calculation of kinetic parameters, the maximum velocity (Vmax), and the Michaelis-Menten constant (KM). ObjectivesThe objectives of this research are: 1) to derive equations without KM for the determination of the Vmax in particular and vice versa; 2) to determine the KM and Vmax with other equations other than the Michaelian equation; and 3) to subject the calculated and extrapolated kinetic parameters to pseudo- statistical remediation where necessary as a test of their viability and usefulness. MethodsThe study was experimental and theoretical. It is supported by the Bernfeld method of enzyme assay. ResultBy graphical means, the Vmax and KM values for galactosidase respectively range between 163 and 185 M/min and between 2.07 and 2.77 mg/L; the range by calculations is 177 and 214 M/min and 2.45 and 3.311 mg/L, subject to pseudo-statistical remediation. Overall, the ranges of Vmax and KM values for alpha-amylase from both the graphical method and calculation are, respectively, 1.095 to 1.018 mM/min and 18.15 to 20.554 g/L. ConclusionThe equations for the determination of the KM and Vmax, which are respectively invariant with respect to each other, were rederived. The initial rates must not be a mixture of both if the true KM and Vmax are of interest. The new pseudo-statistical method for the remediation of error in all measurements, if necessary, is viable, useful, and robust. Graphical abstract O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=82 SRC="FIGDIR/small/537023v1_ufig1.gif" ALT="Figure 1"> View larger version (22K): org.highwire.dtl.DTLVardef@824dfaorg.highwire.dtl.DTLVardef@cd95e5org.highwire.dtl.DTLVardef@12c0771org.highwire.dtl.DTLVardef@19cb86d_HPS_FORMAT_FIGEXP M_FIG O_FLOATNOA:C_FLOATNO Plots where conditions that validate a very high incidence of rQSSA are the case: [ET] is >>[ST]. Plot of v1 to v5 versus [ST]1 to [ST]5 gave equation of linear regression (double reciprocal plot (drp)) such as: y = (0.08x - 0.0002) exp. (+3). A drp plot of all values of v versus all values of [ST] gave a linear regression equation such as: y = (0.08 x + 6 exp. (- 05)) exp. (+3). [Figure 1] stands for a linear regression of v versus [ST] (y = 0.0125x); [Figure 1] stands for a linear regression of 1/v versus 1/[ST]. [ST]nvn-1 - [ST]n-1vn is = zero in all data points. The reciprocal of the intercept gives a very high value (over estimation of the maximum velocity, Vmax (16667 mM/min) and consequently an over estimated Michaelis-Menten constant, KM (KM value is = 106.668 g/L)). C_FIG O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=100 SRC="FIGDIR/small/537023v1_ufig2.gif" ALT="Figure 2"> View larger version (25K): org.highwire.dtl.DTLVardef@10020forg.highwire.dtl.DTLVardef@f75707org.highwire.dtl.DTLVardef@72c474org.highwire.dtl.DTLVardef@bfce2a_HPS_FORMAT_FIGEXP M_FIG O_FLOATNOB:C_FLOATNO Plots where conditions that neither totally validates an incidence of rQSSA nor sQSSA: Some v values are {propto} [ST] while some are not. Plot of all v values versus all [ST] values gave equation of linear regression (double reciprocal plot (drp)) such as: y = (0.6179 x + 0.1973) exp. (+3); the resulting Vmax is = 5.068mM/min and the KM is = 3.132g/L. The linear regression of 1/v versus 1/[ST] gave: y = (0.6682x - 0.0164) exp. (+3) for the plot covering 1/v1 to 1/v5. [Figure 2] stands for a "polynomial regression" of v versus [ST]; [Figure 2] stands for a linear regression of 1/v versus 1/[ST]. [ST]nvn-1 - [ST]n-1vn is = zero where the v values covers v7 to v14; [ST]nvn-1 - [ST]n-1vn is = zero where the v values covers v1 to v6. C_FIG O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=99 SRC="FIGDIR/small/537023v1_ufig3.gif" ALT="Figure 3"> View larger version (24K): org.highwire.dtl.DTLVardef@7856f8org.highwire.dtl.DTLVardef@95ceb1org.highwire.dtl.DTLVardef@134ebb1org.highwire.dtl.DTLVardef@15de459_HPS_FORMAT_FIGEXP M_FIG O_FLOATNOC:C_FLOATNO Plots where conditions that validate an incidence of either rQSSA or sQSSA may be the case: Such conditions are [ST] {approx} [ET]; [ET] < KM. The Vmax value and KM value expected from the regression equation (y = 0.4495 x + 0.2921) exp. (+3) from the plot of 1/v versus 1/[ST] are respectively 3.423 mM/min and 1.54 g/L. [Figure 3] stands for either linear or "polynomial" regression of v versus [ST]: Both plot show R2 that is = 0.9996; [Figure 3] stands for a linear regression (drp) of 1/v versus 1/[ST]. C_FIG O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=87 SRC="FIGDIR/small/537023v1_ufig4.gif" ALT="Figure 4"> View larger version (27K): org.highwire.dtl.DTLVardef@13aa340org.highwire.dtl.DTLVardef@b3c16eorg.highwire.dtl.DTLVardef@1604419org.highwire.dtl.DTLVardef@116dc95_HPS_FORMAT_FIGEXP M_FIG O_FLOATNOD:C_FLOATNO D: Plots where the condition that validate an incidence of sQSSA (or "Henri-Briggs-Haldane-Michaelis- Menten" (HBHMM) equation) may be the case: Such condition is that [ST] is >> [ET]. The Vmax value and KM value expected from the regression equation (y = 0.0449x + 0.0295) exp. (+6) from the plot of 1/v versus 1/[ST] are respectively 33.898 M/min and 1.522 g/L. [Figure 4] stands for a linear regression of 1/v versus 1/[ST]; [Figure 4] stands for a linear regression of v versus [ST]. C_FIG The summary presented in the graphical abstract is primarily intended to remind all and sundry, students and high-ranking scholars in the field, that the issue of QSSA must be reflected in the study of enzyme kinetics because the result of such a study has profound implications for scientific, engineering, and, in particular, medical applications. "To be as imposing as a titanic, does not mean that a titanic-like body is unsinkable". This implies that minor issues that are ignored can ultimately flaw a post-doctoral thesis by high-ranking researchers. Needless to give an example, but what needs to be taken home is that if an enzyme is very active with a given drug (and even food) to be activated, care should be taken to ensure that a low concentration of drug needs to be administered. In the management of diabetics, starchy foods containing resistant starches are recommended for the same reason.