ALGORITHMS FOR PROTEIN FOLDING, PROTEIN‐PROTEIN INTERACTION, MOLECULAR MODELING
OF COMPLEX SYSTEMS, COSMOLOGY, AND HIGH ENERGY PARTICLE INTERACTION MODELING
BASED ON 4‐SPHERE PHYSICS WITH SU(2) ELECTROMAGNETISM
Inventors: Zulfikar Ahmed, Citizen of Bangladesh
Residing in Brooklyn, New York
Feroze Ahmed, Citizen of Bangladesh
Residing in Northern Kentucky
Arjuna Balasingham, Citizen of Australia
Residing in San Francisco, California
Giovanni Barbagelata, Citizen of Italy
Residing in Switzerland
Mikhail Belov, Citizen of United States
Residing in Richland, Washington
David Ben‐Zvi, Citizen of United States
Residing in Austin, Texas
Natalia Brizuela, Citizen of United States
Residing in Buenos Aires, Argentina
Alessio Caldarera, Citizen of Germany
Residing in Tokyo, Japan
Donald Ducote, Citizen of United States
Residing in Brooklyn, NY
Noia Efrat, Citizen of United States
Residing in New York, New York
Tina Ghosh, Citizen of United States
Residing in Washington, DC
Waheed Hussain, Citizen of Canada
Residing in Washington, DC
Everett Kane, Citizen of United States
Residing in New York, NY
Harris Karlin, Citizen of United States
Residing in New York, NY
Jukka Keranen, Citizen of Finland
Residing in Los Angeles, CA
Stephanie T. Lee, Citizen of United States
Residing in New York, NY
Andrew McCollum, Citizen of United States
Residing in Texas
Xenia Protopopescu, Citizen of United States
Residing in New York, NY
Judit Revesz, Citizen of Hungary
Residing in New York, NY
Roberto Samaniego, Citizen of Peru
Residing in Washington, DC
Zachary Stamler, Citizen of United States
Residing in Brooklyn, NY
Vladimir Teichberg, Citizen of United States
Residing in New York, NY
Christopher A. Thorpe, Citizen of United States
Residing in Boston, MA
Thomas Willkens, Citizen of United States
Residing in Brooklyn, NY
BACKGROUND OF THE INVENTION
[001] To be biologically active, proteins must adopt specific folded three‐dimensional, tertiary
structures. Yet the genetic information for the protein specifies only the primary structure, the linear
sequence of amino acids. The primary structure must determine the shape, as proteins refold
spontaneously after being completely unfolded. Traditional approaches of heteropolymer simulation
for biological applications such as the protein folding problem have used mainly used
thermodynamic energy minimization. The present invention consists of algorithms for modeling
kinetics and interactions of biological heteropolymers based on a lattice model and electromagnetic
kinetics. It is known that protein globules are very compact with interior density close to that of
molecular crystals (cf. EI Shakhnovich and AV Finkelstein, Biopolymers 68, 1667(1989).) We
develop electromagnetism as an SU(2) gauge theory on a large 4‐sphere and simulate protein
kinetics geometrically. The natural lattices on a 4‐sphere are described by Delaney‐Dress symbols
(cf. AWM Dress (1987), “Presentations of Discrete Groups, Acting on Simply Connected Manifolds, in
Terms of Parametrized Systems of Coxeter Matrices—A Systematic Approach,” Advances in Math 63,
196‐212.) Since the 4‐dimensional sphere is a quaternionic projective line, spherical tilings can be
described using lattice on flat quaternion spaces.
PHYSICAL THEORY
[001] The present invention consists of a set of algorithms from a nonstandard physical theory. The
theory can be derived from two observational assumptions: first, that the microwave background
radiation is uniformly close to 2.73 K; and second, that protein folding is a robust process in the
sense that if a folded protein is denatured by various means, it refolds to its active configuration
quickly after the denaturing conditions are removed.
[002] The physical theory consists of the following elements: the universe is a round 4‐sphere of
radius approximately 3.88483 x 1016 m where particle physics is described by an SU(3)xSU(2)xSU(2)
gauge theory, all particles follow Hamiltonian classical mechanics, and the physical universe is a 3‐
dimensional submanifold of the 4‐dimensional universe that evolves in time.
[003] DIFFUSION+CMB => BOUNDED UNIVERSE. If particles follow Hamiltonian mechanics, then
large groups of particles can be described as an interacting diffusion. The macroscopic behavior of
interacting diffusions is described by a heat equation of Laplacian with a potential. Assume that the
current distribution of microwave particles in the universe has arisen from diffusion from a point on
a non‐compact Riemannian manifold. In other words, suppose that at time 0, as in the Big Bang
model or the Inflationary universe model, all particles and energy were concentrated at a point, and
now, at time T, there is a distribution with a lower bound of 2.7 K describing the distribution of
microwave photons. If we now assume that the universe is unbounded, or a non‐compact
Riemannian manifold that satisfies Einstein’s field equations, Ricci(X,Y) = const g(X,Y), then the
mathematical results of Peter Li and Shing‐Tung Yau (cf. P Li, S‐T Yau, “On the parabolic kernel of the
Schrodinger operator,” Acta Mathematica, 156 (1986) 153‐201) applies. They prove that the heat
kernel of the Schrodinger operator associated with a large class of reasonable potentials must have
Gaussian upper bounds in the sense that for any point x in the universe, the upper bound as a
function of distance d from the origin of the universe is given by A exp( ‐B d(o,x)2). This function
approaches 0 as the distance gets larger, contradicting the lower bound on observed CMB level. Thus
a constant background radiation is impossible for an unbounded universe. Conversely, if M is a
closed 3 or 4 dimensional manifold, L = Δ + c is a Schrodinger operator acting on smooth functions,
and {ψi} are eigenfunctions of the Laplace‐Beltrami operator on the 4‐sphere of radius R with
eigenvalues λi, then the fundamental solution of the heat equation ∂t F = LF can be written as k(t,x) =
exp(‐ct)Σ exp( ‐λit) ψi2(x). Except for the constant eigenfunction, all others must have a zero, hence
k(t,x) at time t will have an absolute, but achieved lower bound exp(‐ct‐λ0t). In particular, if cosmic
background radiation has occurred by a diffusion with a constant potential, then time from the
beginning of the universe, the total energy at the beginning of the universe, and the spectral gap (the
lowest point in the spectrum) of the universe as a manifold is sufficient to explain and recover the
value of the cosmic background radiation.
[004] The discovery of Schechtman et. al. that aluminum‐manganese has a 5‐fold symmetry, and
therefore not that of crystallographic 3‐dimensional Euclidean groups is evidence that
macroscopically the universe has higher than three dimensions (cf. Schectman, et. al. (1984),
“Metallic Phase with Long‐Range Orientation Order and No Translational Symmetry,” Physical
Review Letters, 1951‐1954. The non‐crystal symmetry groups of quasi‐crystals are icosahedral,
dodecahedral, decagonal, and octagonal point symmetry. Hardy and Silcock have found Al‐Cu‐Li to
form icosahedral quasicrystals, Padazhena and collaborators have found Zn60Mg30Y10 to Bergman
type symmetry. Ishima and collaborators found a phase with 12‐fold symmetry of a Ni‐Cr alloy. A
discrete subgroup D of Rn is a lattice if Rn/D is compact; a set is periodic if S+t=S for some vector t; S
is crystallographic if its periods form a lattice. Orthogonal transformations mapping S to itself must
have finite order in the sense that Ak=I. For n=2 and n=3, the orders can only be 1,2,3,4,6. The
quasicrystals have shown orders 5,10,12 as well. The symmetries of regular polytopes in 4‐
dimensions allow these orders.
[005] Our model assumes that the universe is bounded. Our model further assumes that it is fourdimensional
plus a time dimension. In contrast to Kaluza‐Klein type theories, which are fourdimensional
Ricci flat theories, we choose from compact models of the universe from the class of
Einstein 4‐manifolds. These are manifolds where simplified Einstein field equations of general
relativity, Ricci(X,Y) = λ g(X,Y), holds globally rather than in a three dimensional physical subspace
only. There is an immediate restriction to the value of the constant: in fact, λ = r/4, where r is the
scalar curvature, or trace of the Ricci operator. Since general relativity has been tested rigorously
only on the solar system and binary pulsars, setting the energy‐momentum tensor in the field
equation to zero produces a theory consistent with known experimental results. In this class of
manifolds, we define particle physics via non‐abelian gauge theory – that is, in terms of connections
on principal G‐bundles with self‐dual or anti‐self‐dual curvature form, and G is either SU(3) x SU(2) x
U(1) as in standard particle physics or SU(3) x SU(2) x SU(2) for particle physics with
electromagnetism as an SU(2) gauge theory.
[005] SYMPLECTIC FORM => CLASSICAL MECHANICS. Hamiltonian mechanics is defined for evendimensional
manifolds with a symplectic form in general as follows. A symplectic 2‐form is a
nondegenerate bilinear antisymmetric function w on pairs of tangent vectors on the manifold.
Nondegeneracy means w(X,Y) = 0 for all tangent vectors X and Y only when Y = 0. This defines the
map i mapping tangent vectors X to cotangent vectors by mapping X to w(X,.). On closed manifolds, it
is known that a symplectic form defines a non‐zero element of the second de Rham cohomology
space, that is, the form is closed but not the differential of any differential 1‐form. Hamiltonian
mechanics for an energy function E(x) on the manifold is defined in terms of the Hamiltonian vector
field XE on M defined by i(XE) w = dE. This vector field XE is well‐defined because of the nondegeneracy
of w. When M is compact, the vector field XE integrates to a flow φ that preserved w
because the Lie derivative of w along XE is LXEw = i(XE) dw + d( i(XE) w ) = 0, and XE is tangent to level
sets of E. Thus Hamiltonian flow preserves level sets of E.
[006] COMPACT SYMPLECTIC MANIFOLD => NO CONSERVATION OF ENERGY. Hamiltonian
mechanics of SU(2) electromagnetic particles does not have conservation of energy if the manifold is
symplectic. We prove this using the Hofer‐Zehnder periodic orbit theorem, and the non‐trivial twodimensional
topology of compact symplectic manifolds. Since M has non‐trivial second de Rham
cohomology, we can find a SU(2)‐connection A whose curvature 2‐form has nontrivial cohomology.
Now the curvature 2‐form F is an element of H2(M) with su(2) entries. Choose an embedded noncontractible
2‐sphere dual to an element of H2(M) such that A has nontrivial holonomy on loops on
the 2‐sphere. Now construct an energy function that is constant on the 2‐sphere and apply the
Hofer‐Zehnder theorem on the level set contained on the 2‐sphere. There is a periodic orbit of E on
the level set. If the periodic orbit lies on the 2‐sphere, we have a non‐energy conserving system of
two electromagnetic particles as follows because parallel translation of a point along the closed orbit
by connection A (representing a particle) will increase energy as a function of F on every orbit. In a
later paragraph we will give a construction on the 4‐sphere (which is not symplectic) that allows
Hamiltonian mechanics without breaking conservation of energy.
[007] EINSTEIN CONDITION AUTOMATIC FOR PHYSICAL SUBSPACE. Any three‐dimensional
submanifold of a four‐dimensional Einstein manifold is Einstein because the condition Ricci(X,Y) = λ
g(X,Y) for unit tangent vectors on the ambient space automatically makes such a condition hold for
any submanifold. In particular, the “physical universe” can be described as a time‐indexed 3‐
manifold in the 4‐sphere that contains all massive particles in the universe. Einstein 4‐manifolds
thus admit abstract grand unifications in the sense of consistent unification of gravitation with
particle physics, but not necessarily ones where Hamiltonian mechanics for electromagnetic particles
are energy‐conserving. The general form of Einstein field equations is Ricci(X,Y) ‐ λ g(X,Y) = κ T(X,Y)
where T is the energy‐momentum tensor, but Einstein’s field equations have been tested rigorously
only in the solar system and the binary pulsar, where T = 0, hence the underlying theory of the
present invention is consistent with known observations.
[008] EINSTEIN METRIC AND STABLE CLASSICAL HYDROGEN => SPHERE. If a complete
Riemannian manifold M of dimension n > 1 admits a non‐constant function r such that ∇i∇j r = ‐c2 r
gji, where c is a positive constant, then M is a sphere of radius 1/c in (n+1)‐dimensional Euclidean
space. This is a theorem due to Obata (J. Math Soc. Japan, 14:333‐340, 1962). There are variations of
this type of a theorem that use existence of non‐isometric conformal transformations on a manifold
to characterize the sphere. Since we are considering only compact models that satisfy the Einstein
condition globally, the existence of a one‐parameter group of conformal transformations is sufficient
to conclude that topologically it is isometric to a 4‐sphere by the following theorem (cf. K. Yano and
T. Nagano, “Einstein Spaces Admitting a One‐Parameter Group of Conformal Transformations,”
Annals of Math, 69 (2), 451‐461): Let M be a connected complete Einstein space of dimension n>2
and suppose that a vector field on M generates globally a non‐homothetic conformal transformations.
Then M is isometric to a simply‐connected space of positive constant curvature, i.e., the sphere.
Another rigidity theorem in this direction is a theorem of N H Kuiper, “On conformally‐flat spaces in
the large,” Ann. Math. (2) 50 (1949):916‐924. Now consider an energy function E:M‐>R, and let XE be
its gradient vector field. Then this vector field defines a infinitesimal non‐homothetic conformal
transformations on the space for a classical electromagnetic system of an electron spinning around a
proton. In other words, if we assume the space is Einstein and the electron‐proton pair is a classical
system that is stable (as an SU(2) gauge theory) then we can immediately conclude the space must be
a round 4‐sphere.
[009] The round 4‐sphere of any positive radius is Einstein, non‐symplectic, and admits Hamiltonian
mechanics for all particles with any compact connected Lie group as gauge group, including
G=SU(3)xSU(2)xU(1) or G=SU(3)xSU(2)xSU(2) gauge groups via G‐invariant energy functions on HxH,
which are pairs of quaternions. This is because the four‐sphere is also the quaternionic projective
line, that is, it is described as pairs of quaternions (q1,q2) where for non‐zero quaternions a, (q1,q2) is
identified with (aq1, aq2). Since the second de Rham cohomology space of the four‐sphere is trivial,
all connections on principal G‐bundles are flat or curvature‐less, and there is no obstruction to
inertial motion of particles with any gauge symmetry. In particular, classical mechanics is energyconserving
for particles defined by any non‐abelian gauge theory that describes particles, such as
electromagnetism (U(1) is traditional, and our novel scientific claim will be that it is SU(2)), strong
nuclear force with gauge group SU(3), and weak nuclear force with gauge group SU(2), and indeed
any other compact connected Lie group as gauge group. A novel scientific claim is the consequence
of this analysis, that in fact all particles in the standard model must follow classical mechanics on the
round 4‐sphere of a positive radius. Note that on a symplectic closed 4‐manifold it is possible to
prove that macroscopic electromagnetic systems are unstable by Hamiltonian mechanics.
[010] We will justify the claim of SU(2) as the gauge group of electromagnetism mathematically as
follows. Consider an electron spinning around a proton inertially. Pick a point and consider radial
polar coordinates on the tangent space and consider the inverse image of the center of the proton.
Now parallel translate this vector along the path of the electron (circular). Parallel translation by the
Levi‐Civita connection cannot keep the point constant because the sphere has metric curvature. On
the other hand, there is no holonomy, thus when the orbit completes, the vector returns to its
original point. Projecting this loop from the tangent space to the 4‐sphere produces a loop. This
system describes a larger gauge group than U(1).
[011] Immediate consequences of the theory above are the following: (a) the universe is
deterministic, (b) stability of matter is an immediate consequence of vanishing second de Rham
cohomology. (c) gross description of the early universe is diffusion with a constant potential that
explains the cosmic microwave background level, (d) Hubble’s law can be described by the inertial
spread of particles on the heat flow (the speed is proportional to distance) without reference to an
inflationary universe, (e) discrete energy spectra of the hydrogen atom is a consequence of the
compactness of the universe without additional quantum hypotheses – the equal spacing reflects the
spherical geometry of the universe, (f) all smooth functions in the universe can be decomposed into
eigenfunctions of a Laplace‐Beltrami operator with well‐understood structure (these are pure tones
of the universe). This puts strong quantum restrictions on all classical systems. This follows from
elementary spectral theory of compact operators on Hilbert spaces, as the resolvent of the Laplace‐
Beltrami operator is compact, (g) discrete subgroups of metric‐preserving automorphisms on the 4‐
sphere give rise to Platonic tesselations that form natural grids for algorithms in physics.
[012] DETERMINATION OF UNIVERSAL RADIUS. We provide two methods for the determination of
the radius of the universe. The first method is via Planck’s energy quanta. The equal spacing of the
energy as a function of oscillation frequency E = hν, if these occur as eigenvalues of the Laplace‐
Beltrami operator with a constant potential can only occur for a 4‐dimensional sphere whose radius
is h‐1/2 = 3.88483033 x 1016 m. On a sphere of this radius, the spectrum of the Laplace‐Beltrami
operator is a scaling of the spherical harmonics of the 4‐sphere of unit radius where the scale is
Planck’s quantum. A different method of determining the radius is via general relativity, where the
field equations must be Ricci = r/4, where the r is the scalar curvature and must equal R‐2. The field
equations include an energy‐momentum tensor term, but theory has only been tested in the solar
system and binary pulsars. In 1974 binary pulsar B1913+16 was discovered by Taylor and Hulse.
These were tests of field equations without an energy‐momentum term; the other test was in the
solar system.
[013] LIFTING OF SCHRODINGER OPERATORS. We can lift Schrodinger operators from threedimensional
Euclidean space to the four‐dimensional sphere by restricting the potential to the twosphere
of radius 1/R centered at the origin, and extending the potential of the Schrodinger operator
restricted to this sphere to the 4‐sphere of radius R by the following geometric construction: identify
the 2‐sphere of radius 1/R to a 2‐sphere of radius R of quaternions that is contained in the vector
space spanned by j, k. Identify the 3‐sphere of radius R with SU(2), and use rotations to define the
function on radius R sphere of quaternions. Take two copies of quaternions and use the quaternion
projective line construction to push the potential down to the 4‐sphere of radius R. Under this lifting
construction, the lift of the non‐relativistic Schrodinger operator for the hydrogen atom becomes an
operator of form Δ + c, where c is a constant. The energy spectrum of this operator is different but
numerically close to the energy spectrum of the three‐dimensional Schrodinger equation for
hydrogen.
[014] EXISTENCE OF MAGNETIC MONOPOLES. Theoretical description of magnetic monopoles on a
four‐sphere has been completed in 1978 by Atiyah, Hitchin, and Singer in their seminal paper “Selfduality
and Riemannian Geometry”. In particular, the so‐called instantons, of magnetic monopoles of
SU(2) gauge theory of charge k are in one‐to‐one correspondence with a choice of k pairs of points on
the 4‐sphere modulo the action of SU(2). Thus charge k monopoles form a space of real dimension 8k
– 3.
[015] The present invention is the application of the physical theory described above to describe an
algorithm to determine three dimensional conformations of biological heteropolymers as a function
of a sequence description. In particular, we detail a protein folding algorithms based on SU(2)
electromagnetism, algorithms for interactions between biological molecules such as protein‐protein,
protein‐DNA, simulation algorithms based on a geometric description of biomolecules, and describe
statistical analysis, graphical modeling algorithms, and drug design methods using these algorithms.
SPHERICAL LATTICES
[017] QUATERNIONS LATTICES. We may write the elements of the skew field of quaternions H as
a+bj where a and b are complex, and k = ij. The number j satisfies j2 = ‐1 and jxj‐1=conj(x). Define
vectors spaces over H using left multiplication by elements of H. Inner product is defined as a
function VxV ‐> H say f(x,y) that is H‐linear in y, f(x,y) = conjugate(f(y,x)) and f(x,x) > 0. If an element
g of GL_H(V) preserves the inner product it is called unitary, and the set of all such is denoted U_H(V).
If a basis is known for V, then we call this U(n). The finite subgroups of U_H(1) = SU(2) are conjugate
to one of:
(a) C_m =
(b) D_m =
(c) T =
(d) O =
(e) I =
A lattice is a discrete subgroup of a d‐dimensional Euclidean vector space V. For any lattice there
exist n linearly independent vectors b1, …, bn so that L = Zb1 + … + Zbn. Then n is called the rank of
L, and absolute value of the det(b1,…,bn) is called the volume of L. Let Bd be the d‐dimensional unit
ball. Without loss of generality, assume n=d. The geometry of a lattice is encoded in its Dirichlet‐
Voronoi polytope. This is the polytope that contains all those points lying closer to the origin than all
other points, denoted DV(L). These tile V by translates DV(L) + v for elements v of L. They and their
facets are centrally symmetric. The k‐th successive minima λk(L) is the minimum λ such that λBd
contains at least k linearly independent basis elements of L. A theorem of Minkowski says 2^d/d!
vol(L) < λ1(L) … λd(L) vol(Bd) < 2^d vol(L).
The standard lattice on quaternions are the Lipschitz integers a + bi + cj + dk with integer entries.
Norms on quaternions are calculated as q conj(q), thus have the multiplication property
N(q1q2)=N(q1)N(q2). The Lipschitz integers do not have prime factorization, but if the coefficients
can be either integers or ½‐integers, then the lattice, called Hurwitz integers have unique prime
factorization, unique up to multiplication by unit Hurwitz integers, of which there are 24. Consider
pairs of Hurwitz integers (h1,h2). Under projective identification, this point is identified with (1, h1‐
1h2), and thus Hurwitz rationals on quaternions. Using the formula h‐1 = conj(h)/N(h), the
identification can be made to the point ( N(h1), conj(h1)h2).
The lattice generated by the Hurwitz units with the 96 quaternions obtained by ½(0 + i +/‐ R‐1 j +/‐
R k ) by an even permutation of coordinates produces a group of size 120 in SU(2). The convex hull is
a convex regular 4‐polytope called the 600‐cell.
For a given bond length, our model of molecular movement is the discrete model using the vertices of
the tiling of a 4‐sphere of large radius where relative positions of atomic nuclei are determined by
elements of the icosahedral subgroup of SU(2). The self‐similarity of the tiling allows us to define
levels of tiling where the smallest scale is Planck scale and the largest scale in the order of the
universal radius. At each scale SU(2) electromagnetism dictates kinetics of molecules on this tiling.
Suppose we have a heteropolymer connected chain on this 4‐dimensional lattice. At each time step,
the movement of each protein on the lattice will correspond to one of 120 rotations in a deterministic
manner if the protein is not part of a rigid structure. An amino acid is said to be in a non‐rigid
structure if its only neighbors on the lattice are its sequence partners. Otherwise, it is said to be a
rigid structure.
[017] NATURAL LATTICES ON 4‐SPHERE. A tiling is a subdivision of a plane or a higher‐dimensional
space into closed bounded regions called tiles in such a way that the whole configuration can be
reproduced from a finite assembly of tiles by repeatedly shifting and copying in as many directions as
needed. If D is a group that acts without fixed points on a connected n‐manifold M in such a way that
one can find a D‐equivariant cell decomposition, and this cell‐decomposition allows barycentric
subdivision, then D is said to define a geometric cell complex structure on M. Andreas Dress,
“Presentations of Discrete Groups, Acting on Simply Connected Manifolds, in Terms of Parametrized
Systems of Coxeter Matrices – A Systematic Approach,” Advances in Mathematics 63, 196‐212,
(1987), presented a procedure to describe any tiling of a simply connected manifold in terms of
matrices. The free Coxeter group Σ for the index set I = {0,1,…,n} is the free group generated by σi,
with σi2=1. Suppose C is a set on which Σ acts transitively from the right, and D acts faithfully as a
group of Σ‐automorphisms. The Coxeter matrix M(C) with entries
mij(C) := min { m : C(σiσj)m = C }.
For c an element of C, let Sc be the stabilizer of c, the set of σ that fix c. Let T(c) be the group
T(c) := { τ(σiσj)mij(cτ) τ‐1 : τ ∈Σ }
Now the Coxeter matrix satisfies M(c) = M(σc), and thus T(c) is the same for the whole D‐orbit of c.
There is an exact sequence for each c in and its orbit d, 1 ‐> Σc‐> Σd ‐> D ‐>1, and thus for simplyconnected
manifolds, we have 1 ‐> T(D\C) ‐> Σd ‐> D ‐>1. This presents D as the quotient group of a
free group generated by involutions. A tiling of M can be represented as a D‐equivariant chamber
system on M for which the presentation of D using the exact sequence and the Coxeter matrix
classifies the tiling. This is called a Delaney‐Dress pairs. A Delaney symbol of dimension n is a set C
together with functions s0,…,sn from C into C and functions m0,…,mn1
from C into positive integers
such that the following is true: (a) each element of C can be reached from any other point by
repeatedly applying functions from the set s0,…,sn. (b) si(si(c)) = c, (c) si( sj(c)) = sj( si(c)) when j >
i+1, (d) mi(c) = mi(si(c)) = mi(si+1(c)), (e) define f0i(c) = c and fk+1i(c) = si(si+1(fki(c))). Define rij(c) as
the smallest positive number r such that frij(c) = c where f0ij(c)=c and fk+1ij(c)=sisjfkij(c). Define vij(c) as
the fraction mi(c)/rij(c) if j = i+1 and as 2/rij(c) otherwise.
[018] DESCRIPTION OF NATURAL LATTICES ON 4‐SPHERE. Consider quaternions q = a + bi + cj + dk
written as q = a + bi + j (c ‐ di). Since i^2 = j^2 = k^2 = ‐1, we have exp( i pi ) = exp( j pi ) = exp( k pi) =
‐1. The exponential map exp( q ) = C ( cos t + i sin t ) j ( cos u ‐ i sin u ) for two angles t and u. In
particular, the exponential map maps the imaginary quaternion ball of radius pi onto unit
quaternions (identified with SU(2) and the 3‐sphere). Previously I had given the generators of the
120 element subgroup of SU(2) which includes 5‐fold but not 7‐fold symmetry. These are 24
Hurwitz units plus 96 other units. Consider the <
quaternions, and consider two pairs of lattices on HxH scaled by pi. Now left multiplication by
elements of SU(2) will rotate both lattices in the same way, and rescaling will rescale each the same
way. Thus this defines a lattice on the 4‐sphere. This lattice is preserved by the 120 element
subgroup of SU(2).
Unit octionians form the 7‐sphere, and the projection of HxH restricted to the 7‐sphere produces the
Hopf fibration S^7 ‐> S^4 which is naturally a principal SU(2)‐bundle. This provides us with a
concrete SU(2)‐bundle for electromagnetism.
Change the size of the sphere to radius R, and put lattices with edge size ~Planck by rescaling,
[019] PROTEIN‐FOLDING ALGORITHM REFINEMENT: BEGINNING TEST ON DATA
(a) In a protein, amino acid chains have two peptide bonds. Put the first bond at i so the amino acid
is inscribed in the unit imaginary quaternions and note down the element of 120 element subgroup
where the second peptide bond points toward. Call this element the bend of the amino acid.
(b) Let ABC be the codon from which the protein occurred. The helical structure of RNA assigns
three elements of the group. The sum produces a vector V.
(c) Orthogonal to the bend, there is a set of elements of the subgroup that are rotation choices for
protein folding. Use V determined in (b) to begin rotation on the lattice. If there is an edge joining a
previous element of the protein, stop.
(d) If a rigid substructure forms, stop.
Repeat (b)‐(d) for all amino acids.
Now consider rigid structures on a sublattice with larger edge length and repeat (b)‐(d). This
algorithm has complexity N log(N).
RESULT CHECKS:
(a) Put empirically determined structures on the natural lattice and calculate the sequence of SU(2)
elements that describe these.
(b) Diagnostic checks:
(i) Invariance in codon rotation direction
(ii) Crystalline structure (on natural lattice), etc.
[020] RIGIDITY
Any function (say square‐integrable) on the 4‐sphere can be written as a linear combination c_0 f_0 +
c_1 f_1 + .... c_n f_n + .... where f_i are spherical harmonics. (For 4‐sphere of radius R, rescale these).
Suppose the eigenvalues are e_0, e_1, ... This decomposition holds for any function, and therefore the
energy function of any system of particles can be described as the linear combination of a FIXED set
of eigenfunctions. In particular, ALL systems and not simply microscopic systems, will be
"quantized". One could call this rigidity of spectrum: all systems will share the same set of
eigenvalues with differing integer weights.
This implies that for two gases mixing in the same volume, if the first has energy n_0 e_0 + n_1 e_1 + ...
and the second has energy m_0 e_0 + m_1 e_1 + ... for non‐negative integers n_i, m_i then the
spectrum of their mixture will have the following property.
Suppose they are in a volume V. The total energy of the system is (n_0+m_0) e_0 + (n_1 + m_1) e_1 +
... But one can consider a particle with eigenvalues e_k interacting with one with eigenvalue e_k.
Then the possibilities are limited: if there is a redistribution of energy, then without loss of generality
assume the first particle loses energy. The second one will then gain in the coefficients of e_k,
e_{k+1}, .... Since the coefficients are all non‐negative integers, and the eigenvalues are also integers,
the probabilities that e_k ‐> e_{k+1}, e_k ‐> e_{k+2}, etc. can be calculated.
[021] FUNCTION THEORY. Let d denote the operator d/dx + i d/dy + j d/dz + k d/dw, and dbar be
its conjugate so the Laplacian is d dbar. A 3‐dimensional submanifold of the 4‐sphere divides it into
two disjoint pieces. Any function on the sphere is a linear combination of eigenfunctions and a
harmonic function. A harmonic function is determined by its values on the boundary, as are (I need
to check) eigenfunctions. This allows us in principle to make inferences about what is there in the
non‐physical dimension.
[022] In the application area of prognosis and early detection of diseases, we present an outline of
magnetic monopole descriptions of four‐dimensional extensions of biological systems and present
designs of mechanical devices to monitor such extensions. Mathematical description of SU(2)
magnetic monopoles, the so‐called instantons on the 4‐sphere is well‐known and presented in D.
Freed and K. Uhlenbeck, Instantons and FourManifolds,
MSRI 1991. Simple mechanical devices to
detect four dimensional electromagnetic effects can be constructed based on the following geometric
observation: in SU(2) electromagnetism, both electric and magnetic field lines are circular, while
classical Maxwell’s equations have linear electric field lines. Thus, equipment designed to measure
curvature of EM field lines in three dimensions can verify SU(2) electromagnetism.
[023] In the application area of high energy physics, we present simplifications of calculations based
on classical mechanics rather than more sophisticated methods. The standard model of high energy
physics consists of elementary fermions: leptons described by an SU(2) gauge theory and hadrons
described by an SU(3) gauge theory and interactions described by exchange of vector bosons. On the
4‐sphere, we describe the physics in terms of Hamiltonian dynamics of point particles where
interactions between particles are exchanges of vector boson point particles. This formulation
differs from the description due to Glashow, Weinberg, and Salam (cf. Michael Peskin and Daniel V.
Schroeder, An Introduction to Quantum Field Theory, Perseus 1995) in two ways: first, Hamiltonian
dynamics on 4‐sphere rather than quantum dynamics applies, and interactions can be described via
exchange of bosons or classical descriptions in terms of elastic and non‐elastic collisions of systems.
Non‐abelian gauge theories (cf. C. N. Yang and R. Mills, “Conservation of Isotopic Spins and Isotopic
Gauge Invariance,” Phys. Rev. 96, 191 (1954)) were introduced for work in non‐compact spaces. We
specify their application to 4‐sphere where the quantum character of the theory is a consequence of
the geometry of the space.
DETAILED DESCRIPTION OF THE INVENTION
[024] Interest in Kaluza‐Klein cosmological models, where the universe is assumed to be fourdimensional
have had revived interest recently (cf. Paul Wesson, SpaceTimeMatter:
Modern KaluzaKlein
Theory, World Scientific 1999). These theories make the assumption that the four‐dimensional
universe are Ricci flat. The present invention outlines applications of an alternative model of fourdimensional
universe – a four‐dimensional sphere of fixed radius where the physical universe is an
evolving three‐dimensional submanifold with expansion determined by heat diffusion from a point
source with finite energy.
[025] The invention of the quantum by Planck in 1900 was his solution to the blackbody radiation
problem. An ideal body that absorbs and emits all frequencies of light when heated or cooled is
called a blackbody and serves as an idealization for any radiating material. The radiation emitted by
a blackbody is called a blackbody radiation. Empirical graph with temperature on y‐axis and emitted
light frequency on x‐axis shows curves that are bell‐shaped but skewed to the right. Under the
assumption that the radiation is due to oscillations of electrons in atoms and the introduction of a
quantum of energy, Planck derived the frequency distribution of blackbody as
,
In our model, this law is a consequence not of a quantum particle, but of the spherical geometry of
the universe. If a given electron is in orbit around the nucleus with a radius r, then its stable
oscillations in a 4‐dimensional universe of radius R is constrained to a discrete set of possible
wavelengths. More precisely, one can introduce the 3‐dimensional sphere of radius r and ask for the
standing waves on such a sphere. The pure tones are eigenvalues of an equation of type Δf+cf = λf,
which can only have discrete set of values – the discreteness is true in any bounded universe, but the
arithmetic progression for energy, consistent with energy quanta, are a feature of spheres. In
particular, our model is able to reproduce Planck’s blackbody distribution using classical mechanics
on a sphere.
[026] While a grand unified theory for a gauge theory of SU(3) x SU(2) x U(1) symmetry is consistent
with a 4‐sphere model, the present invention focuses on applications of an SU(3) x SU(2) x SU(2)
symmetry particle physics – where electromagnetism is an SU(2) gauge theory rather than a U(1)
gauge theory. The Yang‐Mills theory for SU(2) electromagnetism is well‐known in the literature of 4‐
dimensional geometry through the work of Simon Donaldson, Michael Atiyah, Karen Uhlenbeck,
Clifford Taubes, John Morgan, Tomasz Mrowka and others (cf. S Donaldson, Geometry of FourManifolds,
Cambridge UP). The present invention presents applications of this mathematical theory
to a model of biological heteropolymers on a grid on the 4‐sphere of fixed radius and discrete models
on such a lattice. First, note that the isometry group of the 4‐sphere is O(5) of orthogonal matrices,
that is, 5x5 matrices that satisfy AtA = I. Discrete subgroups of SO(5) are the 5‐dimensional
crystallographic groups that have been classified, for instance in R. Veysseyre and H. Veysseyre,
“Crystallographic point groups of five‐dimensional space. 1. Their elements and their subgroups,”
(2002) Acta Crystallographica. A58: 429‐433, and “Crystallographic point groups of five‐dimensional
space 2. Their geometric symbols,” A58: 434‐440. The tabulation of 5D crystallographic groups
(describing lattices on which our algorithms apply) have been described in T. Janssen, et. al. “Report
of the Subcommittee on the Nomenclature of n‐Dimensional Crystallography. II. Symbols for
arithmetic crystal classes, Bravais classes and space groups,” Acta Cryst. (2002), A58, 605‐621. The
program developed by W. Plesken and collaborators called CARAT (http://wwwb.math.rwthaachen.
de/carat/) describes the 5‐dimensional groups.
[027] Previous approaches to the protein folding problem have relied on thermodynamic principles
and energy‐minimization (cf. K. A. Dill, S B Ozkan, T R Weikl, J D Chodera, V A Vowlz, “The protein
folding problem: when will it be solved?”, Current Opinion in Structural Biology, 2007, 17:342‐346).
The Shakhnovich group has discovered the energy gap criterion, the nucleation scenario, for
instance, and ab initio all atom simulations of proteins to native states that globally minimize energy.
The fold of a protein can be described statically. The present invention provides a description of the
shape of a protein in terms of a sequence of elements of SU(2), thought of as “turns on 4‐sphere” and
describes an SU(2) electromagnetic mechanism that describes how these rotations are determined
by the RNA for a specific amino acid.
[028] Human lysozyme is a well‐studied protein (cf. C B Anfinsen (1973), “Principles that govern the
folding of protein chains,” Science 181:223‐230, C Tanford, K Aune, A. Ikai (1973), “Kinetics of
unfolding and refolding proteins III. Results for lysozyme.” J. Mol. Biol. 73:185‐197). Our linear
complexity algorithm that describes the shape of lysozyme given the amino acid sequence as input
has the following output. Each rotation is described as a fixed length quaternion (an element of the
Lie algebra of su(2)).
Amino Acid Codon Rotation
M AUG 1/3 + k/3 + j/3
K AAG 1/3 + 1/3 + j/3
A GCC j/3 + i/3 + i/3
L CUU i/3 + k/3 + k/3
I AUC 1/3 + k/3 + i/3
V GUU j/3 + k/3 + k/3
L CUU i/3 + k/3 + k/3
G GGU j/3 + j/3 + k/3
L CUU i/3 + k/3 + k/3
V GUU j/3 + k/3 + k/3
L CUU i/3 + k/3 + k/3
L CUU i/3 + k/3 + k/3
S AGC 1/3 + j/3 + i/3
V GUU j/3 + k/3 + k/3
T ACA 1/3 + i/3 + 1/3
V GUU j/3 + k/3 + k/3
Q CAG i/3 + 1/3 + j/3
G GGU j/3 + j/3 + k/3
K AAG 1/3 + 1/3 + j/3
V GUU j/3 + k/3 + k/3
F UUC k/3 + k/3 + i/3
E GAG j/3 + 1/3 + j/3
R CGU i/3 + j/3 + k/3
C UGU k/3 + j/3 + k/3
E GAG j/3 + 1/3 + j/3
L CUU i/3 + k/3 + k/3
A GCC j/3 + i/3 + i/3
R CGU i/3 + j/3 + k/3
T ACA 1/3 + i/3 + 1/3
L CUU i/3 + k/3 + k/3
K AAG 1/3 + 1/3 + j/3
R CGU i/3 + j/3 + k/3
L CUU i/3 + k/3 + k/3
G GGU j/3 + j/3 + k/3
M AUG 1/3 + k/3 + j/3
D GAU j/3 + 1/3 + k/3
G GGU j/3 + j/3 + k/3
Y UAU k/3 + 1/3 + k/3
R CGU i/3 + j/3 + k/3
G GGU j/3 + j/3 + k/3
I AUC 1/3 + k/3 + i/3
S AGC 1/3 + j/3 + i/3
L CUU i/3 + k/3 + k/3
A GCC j/3 + i/3 + i/3
N AAC 1/3 + 1/3 + i/3
W UGG k/3 + j/3 + j/3
M AUG 1/3 + k/3 + j/3
C UGU k/3 + j/3 + k/3
L CUU i/3 + k/3 + k/3
A GCC j/3 + i/3 + i/3
K AAG 1/3 + 1/3 + j/3
W UGG k/3 + j/3 + j/3
E GAG j/3 + 1/3 + j/3
S AGC 1/3 + j/3 + i/3
G GGU j/3 + j/3 + k/3
Y UAU k/3 + 1/3 + k/3
N AAC 1/3 + 1/3 + i/3
T ACA 1/3 + i/3 + 1/3
R CGU i/3 + j/3 + k/3
A GCC j/3 + i/3 + i/3
T ACA 1/3 + i/3 + 1/3
N AAC 1/3 + 1/3 + i/3
Y UAU k/3 + 1/3 + k/3
N AAC 1/3 + 1/3 + i/3
A GCC j/3 + i/3 + i/3
G GGU j/3 + j/3 + k/3
D GAU j/3 + 1/3 + k/3
R CGU i/3 + j/3 + k/3
S AGC 1/3 + j/3 + i/3
T ACA 1/3 + i/3 + 1/3
D GAU j/3 + 1/3 + k/3
Y UAU k/3 + 1/3 + k/3
G GGU j/3 + j/3 + k/3
I AUC 1/3 + k/3 + i/3
F UUC k/3 + k/3 + i/3
Q CAG i/3 + 1/3 + j/3
I AUC 1/3 + k/3 + i/3
N AAC 1/3 + 1/3 + i/3
S AGC 1/3 + j/3 + i/3
R CGU i/3 + j/3 + k/3
Y UAU k/3 + 1/3 + k/3
W UGG k/3 + j/3 + j/3
C UGU k/3 + j/3 + k/3
N AAC 1/3 + 1/3 + i/3
D GAU j/3 + 1/3 + k/3
G GGU j/3 + j/3 + k/3
K AAG 1/3 + 1/3 + j/3
T ACA 1/3 + i/3 + 1/3
P CCG i/3 + i/3 + j/3
G GGU j/3 + j/3 + k/3
A GCC j/3 + i/3 + i/3
V GUU j/3 + k/3 + k/3
N AAC 1/3 + 1/3 + i/3
A GCC j/3 + i/3 + i/3
C UGU k/3 + j/3 + k/3
H CAC i/3 + 1/3 + i/3
L CUU i/3 + k/3 + k/3
S AGC 1/3 + j/3 + i/3
C UGU k/3 + j/3 + k/3
S AGC 1/3 + j/3 + i/3
A GCC j/3 + i/3 + i/3
L CUU i/3 + k/3 + k/3
L CUU i/3 + k/3 + k/3
Q CAG i/3 + 1/3 + j/3
D GAU j/3 + 1/3 + k/3
N AAC 1/3 + 1/3 + i/3
I AUC 1/3 + k/3 + i/3
A GCC j/3 + i/3 + i/3
D GAU j/3 + 1/3 + k/3
A GCC j/3 + i/3 + i/3
V GUU j/3 + k/3 + k/3
A GCC j/3 + i/3 + i/3
C UGU k/3 + j/3 + k/3
A GCC j/3 + i/3 + i/3
K AAG 1/3 + 1/3 + j/3
R CGU i/3 + j/3 + k/3
V GUU j/3 + k/3 + k/3
V GUU j/3 + k/3 + k/3
R CGU i/3 + j/3 + k/3
D GAU j/3 + 1/3 + k/3
P CCG i/3 + i/3 + j/3
Q CAG i/3 + 1/3 + j/3
G GGU j/3 + j/3 + k/3
I AUC 1/3 + k/3 + i/3
R CGU i/3 + j/3 + k/3
A GCC j/3 + i/3 + i/3
W UGG k/3 + j/3 + j/3
V GUU j/3 + k/3 + k/3
A GCC j/3 + i/3 + i/3
N AAC 1/3 + 1/3 + i/3
R CGU i/3 + j/3 + k/3
N AAC 1/3 + 1/3 + i/3
R CGU i/3 + j/3 + k/3
C UGU k/3 + j/3 + k/3
Q CAG i/3 + 1/3 + j/3
N AAC 1/3 + 1/3 + i/3
R CGU i/3 + j/3 + k/3
D GAU j/3 + 1/3 + k/3
V GUU j/3 + k/3 + k/3
R CGU i/3 + j/3 + k/3
Q CAG i/3 + 1/3 + j/3
Y UAU k/3 + 1/3 + k/3
V GUU j/3 + k/3 + k/3
Q CAG i/3 + 1/3 + j/3
G GGU j/3 + j/3 + k/3
C UGU k/3 + j/3 + k/3
G GGU j/3 + j/3 + k/3
V GUU j/3 + k/3 + k/3
[029] The capacity of proteins to interact specifically with one another underlies our conceptual
understanding of how living systems function. Systems‐level study of specificity in protein‐protein
interactions is complicated by the fact that the cellular environment is crowded and heterogeneous;
interaction pairs may exist at low relative concentrations and thus be presented with many more
opportunities for promiscuous interactions compared with specific interaction possibilities. The
Shakhnovich group has presented a computational model of interacting model proteins immersed in
a mixture of hundreds of different unrelated ones where all of them undergo simulated diffusion and
interaction (see E. Deeds, O. Ashenberg, J. Gerardin, E. Shakhnovich. “Robust protein‐protein
interactions in crowded cellular environments,” Proc. Nat. Acad. Sci. 104 (38), 14952 – 14957). They
have found that there exists a range of temperature [Trand,Tdesign] such that below this range
formation of specific complexes is suppressed by promiscuous interactions, while above it formation
of specific complexes becomes unstable. They explain this in terms of an energy gap between
binding energies of specific complexes and the set of binding energies between randomly associating
proteins. In other words, given a set of proteins in a mixture, their work implies criteria for possible
interactions in terms of binding energy matching between proteins.
[030] Reduced representations of proteins in a cubic lattice were introduced in E. Shakhnovich and
A. Gutin (1990), “Enumeration of all compact conformations of copolymers with random sequence of
links,” 93 (8) J. Chem. Phys., 5967‐71. Their study of compact self‐avoiding conformations of
monomers allowed numerically exact calculation of all thermodynamic functions of a monomer
chain. The present invention presents a non‐Euclidean lattice for the monomers that is compatible
with SU(2) electromagnetism on a large 4‐sphere. The advantages in accuracy and efficiency for
biological monomer chains such as proteins, DNA, and RNA are clear from the fact that such lattices
are natural lattices in our model of the universe.
[025] If T is an isometry of the 4‐sphere in Euclidean 5‐space, then T may be viewed as an
orthogonal linear transformation of E5 restricted to S4. Such isometries map Killing fields to Killing
fields and constant conformal fields (df = (X, .)) into constant conformal fields. Thus if a conformal
field is invariant under T so are its constant and Killing parts.
[026] The present invention provides an algorithm for filtering possible interactions between
proteins based on our geometric representation of proteins as sequences of elements of the Lie
group SU(2) representing relative rotations of each amino acid in the protein with respect to the
previous amino acid in the chain.
[098] The application to protein folding consists of (1) a procedure for automated sequence
extraction from a protein database, MMDB or PDB, (2) a shape determination algorithm based on the
protein sequence using SU(2) electromagnetism, (3) simulation of kinetics based on 4‐sphere
classical mechanics, (4) graphical modeling of the shape and kinetics using computer programs
written in any programming language or package, (5) simulation of interactions between proteins
that use 4DE for any purpose – for example, drug development, medical or chemical research,
taxonomy of diseases, study of post‐translational modifications of proteins, development of
computational technologies based on proteins, or any other commercial application.
[099] Instrumentation for 4D electromagnetism can be based on (a) curvature of EM field lines and
(b) Radon and X‐ray transforms on the sphere, that can be used to infer structure of cross‐sections of
distributions of electromagnetic particles away from the physical subspace.
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