Newly-developed methods from the theory of intrinsically disordered proteins can be applied to the flexible glycan structures that coat cellular surfaces and provide rich channels for biological information transmission. Extension of a mechanistic &#x27;arm-in-sleeve&#x27; model via a nonrigid molecule symmetry analysis leads to expectation of empirical observation of punctuated &#x27;spectral&#x27; classifications in glycan/lectin interaction, parameterized by an appropriate index of glycan frond length or other index of topological complexity, possibly requiring groupoid classifications analogous to quasicrystals

Rodrick Wallace

English

Nature Precedings

Version 7
Empirical predictions of an intrinsically disordered protein
theory approach to glycan/lectin reaction kinetics
Rodrick Wallace
Division of Epidemiology
The New York State Psychiatric Institute ∗
January 2, 2012
Abstract
Newly-developed methods from the theory of intrinsically dis-
ordered proteins can be applied to the flexible glycan struc-
tures that coat cellular surfaces and provide rich channels for
biological information transmission. Extension of a mecha-
nistic ‘arm-in-sleeve’ model via a nonrigid molecule symme-
try analysis leads to expectation of empirical observation of
punctuated ‘spectral’ classifications in glycan/lectin interac-
tion parameterized by an appropriate index of glycan frond
length or other index of topological complexity, possibly re-
quiring groupoid classifications analogous to quasicrystals.
Key Words: group theory, groupoid, information theory,
nonrigid molecule, rate distortion
1 Introduction
Flexible glycan fronds coat the cellular surface and provide a
signaling base for a vast spectrum of interactions with other
biological structures and entities, ranging from tissue part-
ners to pathogens and parasites. The are built from some 7-
10 thousand ‘glycan determinants’, in the sense of Cummings
(2009), making glycan structures far more complicated than
proteins constructed from 20 or so amino acids, intractable
as understanding both protein folding and intrinsically disor-
dered protein (IDP) function remain.
Wallace (2012a) has examined the reducto ad absurdum
implied by application of Tlusty’s elegant rate distortion error
code analysis in the context of so many fundamental building
blocks, finding that the production of flexible glycan fronds at
the cell surface must be regulated by sophisticated processes
of chemical cognition. Here we focus down on the business
end of the matter, where the glycan meets the lectin.
As is well known, while evolutionary conservation is roughly
in the order of genetic code > RNA sequences > primary pro-
tein sequence > metabolic pathways > cellular lipid compo-
sition > surface glycan structures, the information content,
related to structural diversity, is in reverse order.
∗Wallace@nyspi.columbia.edu. Box 47, 1051 Riverside Dr., New
York, NY, 10032
Most glycans, at the information extreme of this series, have
significant freedom of motion in aqueous solution. In addition,
most intrinsic glycan functions are further mediated, not by
a single absolutely required sequence, but by an ensemble of
possible glycan conformations, spanning a continuum that can
carry out functions.
The essential chemical interaction involves an anchored gly-
can surface module and an incoming lectin or other chemical
sensor, a signal exchange that Gabius et al. (2002) character-
ize in terms of a ‘sugar code’.
Glycan flexibility may serve an essential purpose for reac-
tion kinetics, analogous to the increased reaction rates avail-
able to IDP (Gruebele, 2005). The argument adapts recent
formal results in IDP theory (Wallace, 2011a, b, 2012b) to
the analysis of glycan/lectin interactions.
2 Catalysis by mechanical flexibility
The first step is to characterize the reaction between glycan
and lectin in terms of a sequence of intermediate states in
an analog to the free energy funnel of protein folding (Grue-
bele, 2005), essentially an arm-in-sleeve fitting mechanism,
and we are interested in the sequence of intermediate states
leading to the final fit. The first – plausible – assumption
is that, starting with an initial contact state a0, there is a
subsequent identifiable series of glycan/lectin conformations
xn ≡ {a0, a1, ..., an} having the property that a small set is
highly probable, but the vast majority will be in violation
of an inherent conformational chemical grammar and syntax.
The formal requirement is that, given N(n) highly probable
paths of length n, the limit
lim
n→∞
log[N(n)]
n
= H
(1)
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exists and is independent of path. Under such a circum-
stance it is possible to define joint and conditional probabil-
ities P (a0, ..., an), P (an|a0, ..., an−1) such that the appropri-
ate joint and conditional Shannon uncertainties of the form
−∑i Pi log[Pi] exist, and the underlying conditions of the
Shannon-McMillan and Rate Distortion Theorems apply. See
Wallace (2011a) for details. It is also possible to define an av-
erage measure of distortion, say D, between an intermediate
glycan/lectin conformation and the final desired state. Ac-
cording to the Rate Distortion Theorem (Cover and Thomas,
2006), there is a minimum channel capacity, R, such that if
the ‘reaction message’ x is transmitted at a rate less than or
equal to R, then the average distortion of the final config-
uration will be less than D, for any reasonable measure of
distortion. In addition, the relation between R and D will
always be convex, i.e., a reverse-J shaped curve.
For a Gaussian channel having noise of zero mean and vari-
ance σ2 (Cover and Thomas, 2006),
R(D) = (1/2) log[σ2/D], D < σ2
R(D) = 0, D ≥ σ2.
(2)
As Feynman (2000) argues, information is a form of free
energy, and it is easy to construct an ideal machine that can
turn the information within a message into useful work, the
essential characteristic of free energy. Arguing by abduction
from intrinsically disordered protein theory (Wallace, 2011a),
we suppose the channel capacity of the glycan/lectin sleeve
mechanism is capped at some maximum possible value R, and
that the glycan/lectin interaction of equation (1) is character-
ized by some index of topological complexity, λ. That is, we
have a set of limits of the form Hλ depending on interaction
structure. What, under such circumstances, will be the re-
action rate of the glycan/lectin interaction? Following the
intrinsically disordered protein argument of Wallace (2011a),
under limited channel capacity, the probability of Hλ – defin-
ing the reaction rate – will be given by something of the form
P [Hλ] =
exp[−Hλ/µR]∑
β exp[−Hβ/µR]
,
(3)
where µ is a scaling constant and λ, β index topological com-
plexity, so that
log[P (Hλ)] = log[
exp[−Hλ/µR]∑
β exp[−Hβ/µR]
] ≡ C(R)− Hλ
µR
.
(4)
The simplest assumption is that Hλ ∝ λ. Then, using an
integral approximation for the sum,
P [λ] =
exp[−mλ/µR]∫∞
β=0
exp[−mβ/µR] =
m
µR
exp[−mλ
µR
]
(5)
and
log[P (λ)] = log[m/µR]− mλ
µR
.
(6)
This is closely similar to the classic result for protein folding
rate vs. absolute contact order, an index of protein topological
complexity (Gruebele, 2005).
Assuming a Gaussian channel, we can nest equation (2)
within this result to obtain a spectrum of reaction rate re-
lations that now depend on both the topological complexity
of the glycan/lectin sleeve-fitting reaction and on the channel
noise index σ2, as in figure 1.
Consider λ fixed by the chemistry of the glycan/lectin in-
teraction. Following the arguments of Wallace (2011a), the
flexibility of the glycan frond acts as a self-lubricating topo-
logical catalyst that decreases σ2, increasing the reaction rate
with incoming species above what would be expected from a
rigid molecular structure. Something similar might emerge
from equation (4) directly, without need to invoke simplify-
ing assumptions of linear dependence of topological interac-
tion complexity through the index λ that is, in the context of
protein folding, the absolute contact order. Indeed, by such
mechanisms, glycan flexibility may define its attractiveness
to pathogens, suggesting another evolutionary necessity for
rapid changes in glycan structure.
Next, we derive the self-lubrication result using formal
methods, finding the possibility of an observable spectral de-
composition for glycan/lectin interaction.
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Figure 1: Spectrum of relations between log reaction rate and
increasing topological complexity of glycan/lectin interaction
for increasing ‘roughness’ of the ‘reaction rate funnel’ con-
necting glycan determinants with incoming signal species as
measured by the noise σ2 for a Gaussian channel. λ increases
to the right and σ2 increases downward, lowering reaction
rates. Glycan flexibility acts as a self-lubricating topological
catalyst that decreases σ2, increasing the reaction rate with
incoming species above what would be expected from a rigid
molecular structure.
3 Nonrigid molecule theory
The line of argument can be made more precise by invok-
ing same kind of nonrigid molecule argument that Wallace
(2011b, 2012b) has applied to intrinsically disordered pro-
teins: Although we like to think of IDP and glycan/lectin
reactions in classical terms, invoking such metaphors as ‘fly-
casting’ or ‘a snake slithering’, these are actually complicated
processes of quantum chemistry that may benefit from more
formal examination.
Longuet-Higgins (1963), in a classic paper, argues that
The symmetry group of [a nonrigid] molecule is
the set of (i) all feasible permutations of the posi-
tions and spins of identical nuclei and (ii) all feasible
permutation-inversions, which simultaneously invert
the coordinates of all particles in the centre of mass.
As for IDP (Wallace, 2011b, c), we assume it possible to
extend nonrigid molecule group theory to whip-like fronds an-
chored at one end via wreath, semidirect, or other products
over a set of finite and/or compact groups (e.g., Balasub-
ramanian, 1980, 2004), or their groupoid generalizations, as
now common in stereochemistry (Wallace, 2011c and cited
references). These are taken as parameterized by an index of
‘frond length’ L which might simply be the number of exposed
monosaccharides. In general, the number of group/groupoid
elements can be expected to grow exponentially with L, typ-
ically as
∑
Πj |Gj ||Aj |L, where |Gk| and |Ak| are the size, in
an appropriate sense, of symmetry groups Gk and Ak. See
the Balasubraminian references for details.
Kahraman (2009) argues that the observed ‘sloppiness’ of
large lock/small key molecular reaction dynamics suggests
that binding site symmetry may be greater than binding lig-
and symmetries. Thus binding ligands may be expected to in-
volve dual, mirror subgroups/groupoids of the anchored non-
rigid group/groupoid symmetries of the glycan kelp frond.
Thus the argument becomes
Increasing L, |G|, |A| → more flexibility→ greatly enlarged
binding site nonrigid symmetry group/groupoid→ more sub-
groups/subtilings of possible binding sites for ligand attach-
ment.
This is very precisely the ‘topological catalysis’ that de-
creases the ‘friction’ due to σ2 in the self-lubrication argu-
ment, reexpressed in the language of nonrigid molecular sym-
metry.
Glycan ‘sequence flexibility’, i.e., the way in which intrinsic
glycan functions can be carried out by an ensemble of pos-
sible glycan structures, a continuum, mirrors, somewhat, the
dynamic matching of the fuzzy-key-and-lock mechanisms sug-
gested by Tompa and Fuxreiter (2008). Taking the approach
of Wallace (2012b), this can be addressed by supposing that
the duality between a subgroup/subgroupoid of the glycan
frond and the binding ligand site can be expressed as
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Bα = CβDγ
(7)
where Bα is a subgroup/groupoid (or set of them) of the gly-
can nonrigid symmetry group or groupoid, Dγ a similar struc-
ture of the binding ligand, and Cβ is an appropriate inversion
operation or set of them that represents static or dynamic
matching of the fuzzy key to the fuzzy lock. The fuzzyness,
however, now extends to sequence replacement as well as vari-
ations in position, a far more complicated matter that may re-
quire an explicit extension of theory using groupoid methods
and groupoid representations, as has been the case for stereo-
chemistry (again, Wallace, 2011c and references). Groupoids
are, in a sense, local symmetry structures that characterize
the partial symmetries of finite tilings, quasicrystals, and the
like.
4 Extending the model
An essential outcome of this approach is that the glycan
symmetries, and their associated dynamics, should be highly
punctuated in the parameter L that indexes final glycan topo-
logical form, and this effect should be observable.
For large L we apply a statistical mechanics analog for
which we can use Landau’s spontaneous symmetry break-
ing/lifting approach via a Morse theory argument, following
Wallace (2012b). Typically, very many Morse functions are
possible under a given circumstance, and we construct what
is perhaps the simplest using group representations.
Taking an appropriate group representation in a particular
matrix algebra, we can construct a ‘pseudo probability’ P for
nonrigid group element ω as
P[ω] = exp[−|χω|/κL]∑
ν exp[−|χν |/κL]
,
(8)
where χφ is the character of the group element φ in that
representation, i.e., the trace of the matrix assigned to φ, and
|...| is the norm of the character, a real number. For systems
that include compact groups, the sum may be a generalized
integral.
The central idea is that F in the construct
exp[−F/κL] =
∑
ν
exp[−|χν |/κL]
(9)
is a Morse Function in the temperature-analog L to which
we can apply Landau’s spontaneous symmetry breaking ar-
guments (Pettini, 2007; Landau and Lifshitz, 2007), leading to
the expectation of empirically-observable highly punctuated
structure and reaction dynamics in the index L that are the
analog to phase transitions in ‘simple’ physical systems: Lan-
dau’s central insight was that, for many physical phenomena,
raising the temperature would make accessible higher energy
states of the system Hamiltonian, the quantum mechanical en-
ergy operator, and that the inherent symmetry changes would
necessarily be punctuated. Here we focus directly on a Morse
Function constructed from those symmetries, and rely on the
robustness of the underlying mathematics to carry through,
basically an empirical question.
5 Discussion and conclusions
We have extended the ‘snake’ model of intrinsically disor-
dered proteins (Wallace, 2011a), as supplemented by a non-
rigid molecule symmetry approach (Wallace, 2012b), to the
flexible glycan/lectin system that coats the surface of the cell
and provides an exceedingly rich channel for the transmis-
sion of biological information. A quantum chemistry context
is implicit in the application of nonrigid molecule theory, al-
though local tiling-matching may be characterized more by
groupoid than group symmetries, as is the case for quasicrys-
tals and groupoid stereochemistry (Wallace, 2011c and refer-
ences therein).
The essential empirical implication is that sufficient search
should reveal punctuated spectral transitions in glycan/lectin
interaction that can be indexed by an appropriate measure of
glycan ‘length’ – i.e., topological complexity – analogous to
a generalized temperature. These spectral structures are in
addition to the ‘developmental’ spectra predicted by Wallace
(2012a) to characterize the glycan surface fronds themselves.
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2011a, Structure and dynamics of the `protein folding code' inferred using Tlusty's topological reate distortion approach,

2011b, Multifunction moonlighting and intrinsically disordered proteins: Information catalysis, nonrigid molecule symmetries, and the `logic gate' spectrum, Comptes Rendus Chimie,

2011c, On the evolution of homochirality,

2012a, Extending Tlusty's rate distortion index theorem method to the glycome: Do even `low level' biochemical phenomena require sophisticated cognitive paradigms?

2012b, Spontaneous symmetry breaking in a nonrigid molecule approach to intrinsically disordered proteins, Molecular BioSystems, 8:374-377. N a t u r e P r e c e d i n g s : h d l : / n p r e

A model for the emergence of the genetic code as a transition in a noisy information channel,

Consciousness: A Mathematical Treatment of the Global Neuronal Workspace Model,

Downhill protein folding: evolution meets physics,

Fuzzy complexes: polymorphism and structural disorder in protein-protein interactions,

Geometry and Topology in Hamiltonian Dynamics,

Lectures on Computation, Westview, New York. N a t u r e P r e c e d i n g s : h d l :

Principles that govern the folding of protein chains,

Relativistic double group spinor representations of nonrigid molecules,

Statistical Physics, Part I, Elsevier,

The geometry and physiochemistry of protein binding sites and ligands and their detection in electron density maps, PhD Dissertation,

The repertoire of glycan determinants in the human glycome,

The sugar code: functional lectinomics,

The symmetry groups of nonrigid molecules as generalized wreath products and their representations,

The symmetry groups of nonrigid molecules,

Empirical predictions of an intrinsically disordered protein theory approach to glycan/lectin reaction kinetics

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Empirical predictions of an intrinsically disordered protein theory approach to glycan/lectin reaction kinetics

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