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phylogenetic motif Any motif conserved Functions on P with Poisson bracket form a
over biological species. See also biochemical, Lie algebra. There exists a Lie algebra homo-
chemical, dynamical, functional, kinetic, mech- morphism between the Lie algebra of functions
anistic, regulatory, thermodynamic, and topo- on P and vector fields, given by
logical motives.
H → X (f ) ={H, f }.
H
physicalquantity(measurablequantity) An
attribute of a phenomenon, body, or substance It is an isomorphism when constants are quo-
that may be distinguished qualitatively and deter- tiented out of functions.
mined quantitatively. More generally, let P be a smooth manifold
and C (P ) the space of smooth functions. A
∞
Poisson bracket or Poisson structure on P is an
planar graph A graph that can be drawn in
∞
∞
∞
the plane with no edges crossing. operation { , } : C (P ) × C (P ) → C (P )
satisfying the following
plasmid An extrachromosomal genetic ele- (i.) {F, G} is real, bilinear in F and G,
ment consisting generally of a circular duplex
(ii.) {F, G, }=−{G, F}, skew symmetric,
of DNA which can replicate independently of
chromosomal DNA. R-plasmids are responsible (iii.) {{F, G},H}+{{H, F},G}+{{G, H},
for the mutual transfer of antibiotic resistance F}= 0, Jacobi identity,
among microbes. Plasmids are used as vectors (iv.) {FG, H}= F{G, H}+{F, H}G, Leib-
for cloning DNA in bacteria or yeast host cells. niz rule.

ploidy A term indicating the number of sets n
Poisson equation Let U ⊂ R open and u :
of chromosomes present in an organism, e.g., U × R → R. The (nonlinear) Poisson equation
haploid (one) or diploid (two).
for u is
− u = f(x).
Poisson-Boltzmann equation A mathemat-
ical model for the ionic gas (plasma) or ionic
solution. The nonlinear equation is established Poisson manifold A smooth manifold P
based on two physical laws: the Poisson equation with a Poisson bracket { , }. Examples: Sym-
relating the electric potential to charge distribu- plectic manifolds are Poisson manifolds, where
tion, and the Boltzmann law relating the charge the Poisson bracket is defined by
distribution to electric potential. It has been
shown that this equation is a good model for {F, G}(x) = ω(x)(X (x), X (x)), x ∈ M
G
F
many applications even though it suffers from
some thermodynamic inconsistency. The lin- where X is the Hamiltonian vector field of F.
F
earized equation and its solution is known as
Debye-H¨uckel theory. The one-dimensional ver-
polar coordinates The parameterization of
sion of the nonlinear equation can be applied 2
the plane R as X = (r, θ), where r> 0 is the
to modeling the charge distribution near a flat,
absolute value |X| and θ is the angle (in radians)
charged membrane; this is known as Guy-
between the horizontalaxis and the segmentfrom
Chapman equation.
O to X.
Poisson bracket Let (P, ω) be a symplectic
A
manifold, ξ a system of canonical coordinates pole Acomplexnumberz isapoleofafunc-
0
and f , g two real functions on P. Then the Pois- tion f(z) if f(z) is analytic in 0 < |z − z | <",
0
son bracket of f and g is defined as the function for some "> 0 and not analytic at z ,but
0
n
(z − z ) f(z) is analytic at z , for some positive
0
0
−1 AB
{f, g}= ω (df, dg) = ω (∂ f )(∂ g). integer n.
A B
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