How does one calculate K_M and v_max from the Lineweaver-Burk (Double-Reciprocal) plot?

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    The equation for the Lineweaver-Burk plot is gotten by doing as the alternative name suggests… taking the reciprocal.
    GENERAL REACTION
    ##mathbf(E + S stackrel(k_1)(rightleftharpoons) ES stackrel(k_2)(->) E + P)##
    ##color(white)(aaaaaa)^(mathbf(k_(-1)))##
    LINEWEAVER-BURK PLOTS WITHOUT INHIBITOR
    ##mathbf(v_0 = (v_max[S])/(K_M + [S]))##
    where ##v_max = k_2[E]_total## ##k_2## is the observed rate constant for the conversion of the enzyme-substrate complex to the free enzyme and the product and ##[E]_total## is the total concentration of the enzyme (free complexed whatever).
    So naturally you reciprocate as follows:
    ##1/(v_0) = (K_M + [S])/(v_max[S])##
    ##1/(v_0) = (K_M)/(v_max[S]) + cancel([S])/(v_maxcancel([S]))##
    ##color(blue)(1/(v_0) = (K_M)/(v_max)1/([S]) + 1/(v_max))##
    Once you plot ##1/v_0## vs. ##1/[S]## you have a slope of ##K_M/v_(max)## and a y-intercept of ##1/v_(max)##. You can solve it from there.
    Of course this is assuming that there is no inhibitor. If there is an inhibitor then you can have either of the following reactions:
    ENZYME INHIBITION
    ##mathbf(E + I rightleftharpoons EI)##
    ##K_I = ([E][I])/([EI])##
    where ##K_I## is the dissociation constant for the ##EI## complex into the free enzyme and the inhibitor.
    ##mathbf(ES + I rightleftharpoons ESI)##
    ##K_I’ = ([ES][I])/([ESI])##
    where ##K_I’## is the dissociation constant for the ##ESI## complex into the ##ES## complex and the inhibitor.
    The resultant Lineweaver-Burk equations are still basically identical other than the fact that now we’d use ##K_M^app## and ##v_max^app## which have different definitions in each case and are used in place of ##K_M## and ##v_max## respectively.
    LINEWEAVER-BURK EQUATIONS FOR INHIBITION
    Competitive Inhibition (binds only to free enzyme):
    ##color(blue)(1/(v_0) = (alphaK_M)/(v_max)1/([S]) + 1/(v_max))##
    where ##alpha = 1 + ([I])/(K_I)##.
    Note that here ##K_M^app = alphaK_M## and ##v_max^app = v_max##.
    Uncompetitive Inhibition (binds only to ##ES## complex):
    ##color(blue)(1/(v_0) = (K_M)/(v_max)1/([S]) + alpha/(v_max))##
    where ##alpha = 1 + ([I])/(K_I)##.
    Note that here ##K_M^app = (K_M)/(alpha)## and ##v_max^app = (v_max)/(alpha)##.
    Pure Non-Competitive Inhibition (binds onto enzyme and ##ES## complex with equal affinity):
    ##color(blue)(1/(v_0) = (alphaK_M)/(v_max)1/([S]) + (alpha)/(v_max))##
    where ##alpha = 1 + ([I])/(K_I)##.
    Note that here ##K_M^app = K_M## and ##v_max^app = (v_max)/(alpha)##.
    Mixed Non-Competitive Inhibition (binds onto enzyme and ##ES## complex with different affinities):
    ##color(blue)(1/(v_0) = (alphaK_M)/(v_max)1/([S]) + (alpha’)/(v_max))##
    where ##alpha = 1 + ([I])/(K_I)## and ##alpha’ = 1 + ([I])/(K_I’)##.
    Note that here ##K_M^app = (alphaK_M)/(alpha’)## and ##v_max^app = (v_max)/(alpha’)##.

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