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primitive change One of the conceptually reversible reaction. See also dextralateral, direc-
simpler molecular changes into which an ele- tion, dynamic equilibrium, formal reaction equa-
mentary reaction can be notionally dissected. tion, microscopic reversibility, rate constant,
Such changes include bond rupture, bond for- reversibility, sinistralateral, and substrate.
mation, internal rotation, change of bond length
projectable vector field A projectable vec-
or bond angle, bond migration, redistribution of
tor field is a vector field on abundle(B,M,π; F)
charge, etc.
inducing by projection a vector field on the base
The concept of primitive change is helpful
manifold M. Locally, it is expressed as
in the detailed verbal description of elementary
i
µ
reactions, but a primitive change does not repre- \ = ξ (x)∂ + ξ (x, y)∂ i
µ
sent a process that is by itself necessarily observ- µ i
able as a component of an elementary reaction. where (x ; y ) are fibered coordinates. The flow
of a projectable vector field is formed by fibered
principal bundle A bundle (P, M, p; G) automorphisms. Vector fields that are not pro-
with a Lie group G as standard fiber and transi- jectable do not preserve the bundle structure.
tion functions valued in the same G and acting
protein folding The dynamical process of
on G by means of left translations L : G → obtaining its structure (see protein structure),
g
G : h → g · h. On principal bundles a global
under appropriate conditions such as tempera-
and canonical right action R of G is defined by ture, pH, etc., from an arbitrary initial structure.
g
In test tubes, it is known experimentally that
R : P → P : (x, h) → (x, h · g).
g
many proteins can reach their respective, almost
This action is free, transitive on fibers and verti- unique three-dimensional structures. How this
cal (i.e., it is formed by vertical automorphisms), process works is still not completely known,
and it completely characterizes the principal even though it is generally accepted that the pro-
bundle. An object is called equivariant if it pre- cess is governed by the intramolecular interac-
serves the canonical right action R . tions between atoms and interaction between the
g
On a principal bundle there is a one-to-one molecule and water in which the process occurs.
correspondence between local trivializations and
protein structure A protein molecule has
local sections. Hence, a principal bundle is triv- a well-defined three-dimensional structure in
ial if and only if it allows a global section.
terms of the relative spatial coordinates of all
its atoms. Most of our current knowledge about
principal connection A connection over a
the protein structures are derived from labora-
principal bundle (P, M, p; G) which preserves
the canonical right action, i.e., tory determinations of the structures of many
proteins using the methods of x-ray crystallogra-
phy or nuclear magnetic resonance spectroscopy.
TR (H ) = H p·g .
p
g
The former requires a protein to form a crystal,
In physics it is also called a gauge field. while the latter can obtain the structure of a pro-
tein in aqueous solution. No reliable computa-
product In chemistry and biochemistry, the tional method exists yet for obtaining the spatial
molecular species yielded by a reaction running structure of a protein from its chemical struc-
in a particular direction. ture (known as its amino acid sequence in bio-
Comment: The distinction is between the chemical literature).
molecular result of a reaction and where a com-
pseudodifferential operator A pseudo-
pound is represented in the arbitrarily written for-
differential operator P of order k on a com-
mal reaction equation. When a reaction occurs
pact manifold M is locally of the form: for any
in a particular direction (either forward or back-
u ∈ C (M)
∞
ward), the substrates and products are known, c
and this identification of molecules with chem-
P u(x) =
ical roles is independent of where the molecules −n i(x−y)·ξ
(2π) e a(x, y, ξ) × u(y)dydξ,
appear in the formal equation representing the

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