Page 76 -

kinetic energy, E k Energy of motion. For
2
a body of mass m, E = mv /2, where v is the
k
speed.
K kinetic equation A balanced reaction equa-
tion which describes an elementary step in
the kinetic sequence of an overall biochemical
Karle-Hauptman method In x-ray crystal- reaction among unambiguously identified reac-
lography, the diffraction pattern is essentially tants. There is no constraint on the number of
the Fourier transform of the periodic crystal participating species. Equally, a set of such equa-
structure which has both an amplitude and a tions which can be combined by any allowed
phase. In practice, the phase information cannot operation or combination of operations to pro-
be obtained; hence, one has to invert the Fourier duce kinetics identical to those measured for the
transform without the phase and, hence, the algo- overall biochemical reaction.
rithm is not unique. J. Karle and H. Haupt- Comment: The requirement that the equa-
man developed a system of inequalities which the tion be elementary ensures that its order will be
amplitudes have to satisfy. Using these inequal- the sum of the molecularities of the reactants
ities as constraints, one can uniquely determine (otherwise, it will be the reactants’ activities).
the inverse Fourier transform if the number of Kinetic sequences here considered correspond
atoms in a unit cell is sufficiently small. The to the common word descriptors (sequential,
inequalities are based on the fact that the elec- ping-pong, etc.). The requirement for identified
tron density is a positive function of which the species distinguishes among different proteins
diffraction is the Fourier transform (cf. J. Karle, which catalyze the “same” reactions but at differ-
J. Chem. Inf. Comput. Sci., 34, 381, 1994). ent rates. Since most kinetic sequences consist
of more than one elementary step, the equations
killing field On a Riemannian manifold
will usually occur in groups. Since not all reac-
(M, g) a vector field X such that the Lie deriva- tions are sequential, the definition provides that
tive L g = 0.
X they can be combined by operators other than
summation, such as Boolean operators or coeffi-
killing vector Over a pseudo-Riemannian
µ cients representing the frequency of a particular
manifold (M, g),a vector field ξ = ξ (x) ∂ ∈ type of event in a population of simultaneously
µ
X(M) such that the Lie derivative $ g vanishes.
ξ
α occurring events.
} are the Christoffel symbols of the metric
If { βµ g
g, ξ is a killing vector if and only if it satisfies
kinetic equivalence Referring to two reac-
the killing equation:
tion schemes which imply the same rate law.For
∇ ξ +∇ ξ = 0 example, consider the two schemes (i.) and (ii.)
ν µ
µ ν
for the formation of C from A:
λ λ
} ξ
where ξ = g ξ and ∇ ξ = ∂ ξ −{ νµ g λ k 1 ,OH − k 2
µ ν
µ ν
λν
ν
Accordingly, killing vectors are infinitesimal (i.) A −→ B −→ C
←−
k −1 ,OH −
generators of isometries of g. A killing vec- providing that B does not accumulate as a reac-
tor is uniquely determined once one specifies its tion intermediate.
µ
value ξ (x ) at a point x ∈ M together with
0 0 −
µ
1 2
its derivatives ∂ ξ (x ). (In fact, by deriving of d[C] = k k [A][OH ] (1)
0
ν
−
the killing equation one obtains a Cauchy prob- dt k + k [OH ]
2
−1
lem which uniquely determines the component k 2
µ
k 1
functions ξ .) On a manifold M of dimension m (ii.) A −→ B OH − C
−→
←−
one can have at most m(m + 1)/2 killing vec- k −1
Providing that B does not accumulate as a reac-
tors. The manifolds with exactly m(m + 1)/2
tion intermediate:
killing vectors are called maximally symmetric;
−
e.g., the plane, the sphere, and the hypersphere d[C] k k [A][OH ]
1 2
= (2)
are maximally symmetric. dt k + k [OH ]
−
−1 2
© 2003 by CRC Press LLC