Untangling Complex Systems, page 36
described for the first time by Pauling in 1935 for explaining the positive cooperativity of hemoglobin.
It was then taken up by D. E. Koshland, G. Némethy and D. Filmer (the so-called KNF model) in 1966.
Its basic assumptions are that the two conformational states ( T and R) are available to each subunit; only the subunit to which the ligand is bound changes its conformation, and the ligand-induced conformational change in one subunit alters its interactions with neighboring subunits. It is a “sequential”
model. A ligand binding induces a change in conformation of an individual subunit without all the rest
of the subunits switching conformations. KNF model is represented by the structures enclosed by the
grey patch, across the diagonal in Figure 7.4, wherein “hybrid” R/T multimers exist.
Both the Pauling-KNF and the MWC models are limiting cases of a more complicated situation.
In fact, proteins are relatively soft polymers and, consequently, have significant structural fluctuations
at room temperature. The static view of the structure of molecules has to be replaced by a dynamic
picture (Motlagh et al. 2014). Allosteric proteins and biopolymers, in general, are made of semi-rigid
domains and subunits, which can move relative to each other, so that “everything that living things
172
Untangling Complex Systems
L
L
L
L
L
L L
L L
L L
L L
L L
L L
L L
L L
L L
L L
L
L
L
L
L
L L
L L
L L
L L
L L
L L
L L
L L
L L
L L
FIGURE 7.4 Models for allostery in the case of a four-subunit enzyme. The squares and the circles designate
the T and R conformations of the subunits, respectively. L is the ligand. The Pauling-KNF model, represented by the elements along the diagonal, considers tertiary structural changes; the MWC model, represented by the
elements of the right and left columns, considers only quaternary structural changes.
do can be understood in terms of jigglings and wigglings of atoms,” as Richard Feynman foresaw
in 1963 in his Lectures in Physics (Feynman et al. 1963). Recent advances in structural biology
indicate that the categories by which protein structures are described (primary, secondary, tertiary,
and quaternary hierarchical structural levels) do not encompass the full spectrum. Proteins exist
as complex statistical ensembles of conformers, and they unfold and fold continuously in localized
regions (Goodey and Benkovic 2008). Many proteins or regions of proteins are structurally disor-
dered in their native, functional state and they cannot be adequately described by a single equilib-
rium 3D structure, but as a multiplicity of states (Tompa 2002). This multiplicity of states has been
named as “super-tertiary structure” (Tompa 2012) in the case of single proteins and as “fuzziness”
in the case of protein-protein complexes (Tompa and Fuxreiter 2007). The benefits are flexibility
and adaptability in recognition, binding promiscuity, all features that confer an exceptionally plastic
behavior in response to the need for the cell. The study of super-tertiary structure and fuzziness can
be accomplished by NMR, X-ray crystallography, electron microscopy, atomic force microscopy,
light-scattering, fluorescence spectroscopy complemented by advanced computational modeling, and
data analysis. In time-resolved spectroscopies, the Maximum Entropy Method (MEM) is a power-
ful data analysis technique for detecting the multiplicity of molecular states and their dynamics. For
example, in an ensemble of hemoglobins, each protein can display its dynamics of oxygen binding.
As a consequence, the protein ensemble should be properly described by a distribution function of
rate constants f( k). The term f( k) dk represents the fraction of molecules reacting with the rate k ± dk.
Information on the rate constant spectrum can be extracted from experiments by recording the kinet-
ics N( t) and expressing it as a weighted infinite sum of exponentials (Brochon 1994):
∞
N t
f k e− kt
( ) =
( ) dk
∫
. [7.11]
0
The Emergence of Temporal Order within a Living Being
173
The recovery of the shapes of the distributions of kinetic constants f( k) is a nontrivial problem in
numerical analysis. It is known as the inversion of the Laplace transform, which is notoriously
ill-condition (see Appendix B). Methods involving the minimization of chi-squared ( χ 2) are commonly used. It is routine to fit N( t) to one- or two-exponential functions by iterative least-squares techniques. When two exponentials fail to fit, three are invoked. When also a three-exponential
function fails, one is tempted to try four. Three and four exponentials will provide a good fit with
visually random residuals and a good χ 2. However, the parameters obtained from these fits are, in
general, meaningless if the data arise from a true distribution of kinetic constants. In these cases, it
is suitable to reckon on the MEM. MEM gives the most probable distribution of exponentials, f( k),
fitting N( t) according to equation [7.11], by maximizing the function
Q = λ S − C, [7.12]
where S is the Shannon-Jaynes5 entropy function
f k
S =
f
∑ ( k
i
)
( )
( )
i − mi − f ki log
[7.13]
m
i
i
formulated for a large but discrete set of rate constants ( i = 1…, n with n → ∞), where the ensemble of m values is a “prior” distribution used to incorporate prior knowledge into the solution;
i
λ is a
Lagrange multiplier and C, in the case of normally distributed noise, takes the familiar form
2
2
( N
)
calc − N
C
exp
= χ =
, [7.14]
σ 2
where σ2 is the variance of data points.
One example of the application of MEM is shown in Figure 7.5 reporting two-lifetime dis-
tributions obtained to fit the luminescence decay kinetics of two distinct samples of tryptophan.
NH2
40
COOH
N
∗ 100]
H
/∑f j[fj 20
0
3
6
9
τ (ns)
FIGURE 7.5 Fluorescence lifetime distributions for Tryptophan as a single molecule (grey data) and bound
to Albumin (black data).
5 Edwin Thompson Jaynes (1922, Waterloo, Iowa–1998) wrote extensively on statistical mechanics and foundations of probability and statistical inference. In 1957, Jaynes proposed the idea of maximum entropy for inferring distributions from data, starting from the observation that the statistical entropy looks the same as Shannon’s information uncertainty.
According to Jaynes, the “least biased” guess at distribution is to find the distribution of highest entropy.
174
Untangling Complex Systems
The distribution in grey is relative to isolated tryptophan molecules in phosphate buffer solution;
the distribution in black is relative to the tryptophan as a residue bound to the protein Albumin.
The fluorophore tryptophan exhibits a much broader distribution of lifetimes within the macromol-
ecule since it experiences many more micro-environment, due to the many conformers of Albumin
(Gentili et al. 2008).
7.2.4 glycolysis
The nonlinear behavior of allosteric enzymes can give rise to oscillations. An example appears
in glycolysis. Glycolysis is a fundamental metabolic process because it produces chemical energy
under the shape of ATP and NADH (Nicotinamide Adenine Dinucleotide) by breaking down glu-
cose (C H O ) into pyruvate (C H O −). A key role in glycolysis is played by the allosteric enzyme
6
12
6
3
3
3
Phosphofructokinase (PFK), having distinct regulatory and catalytic sites and exhibiting hetero-
tropic interactions (Schirmer and Evans 1990). PFK is involved in the step of the glycolysis where a
phosphate group is transferred from ATP to fructose-6-phosphate (F-6-P, see Figure 7.6).
ATP can bind to both the catalytic and the regulatory site of PFK. When it binds to the catalytic
site, it plays as a substrate and phosphorylates F-6-P. On the other hand, when the concentration of
the cellular energy vector ATP is high, the cell does not need to convert further glucose into pyru-
vate. Therefore, a regulatory process of negative feedback takes place: ATP binds to the regulatory
site of PFK and lowers the affinity of the enzyme for its other substrate, i.e., F-6-P. ATP inhibits
the reaction shown in Figure 7.6. Even fructose-1,6-bisphosphate (F-1,6-bP), i.e., the product of the reaction, binds to the effector site of PFK. Unlike ATP, F-1,6-bP actives the enzyme. F-1,6-bP
participates in an autocatalytic step. The autocatalysis is evident if we look at the elementary steps
shown in Figure 7.7. In such elementary steps, PFK(ATP) represents the complex between PFK with
ATP in its active site. PFK[(ATP)+(F-6-bP)] is the complex of PFK that has both ATP and F-6-P in
its active site. (F-1,6-bP)PFK[(ATP)+(F-6-P)] is the enzyme having ATP and F-6-P in its active site
and the product (F-1,6-bP) in its regulatory site.
Such autocatalysis favors oscillations that have been observed to have periods of the order of
several minutes (Das and Busse 1985). The steps involving PFK are represented schematically in
Figure 7.8.
NH
NH
2
2
N
N
O
O
O
N
O
O
N
O–
P O
P
O P O CH
N
N
N
2
N
HO
P O
P
O CH2 O
O–
O–
O–
O
O–
O–
OH
OH
OH
OH
(ATP)
(ADP)
+ PFK
+
+
O
O–
P
–2O
OH
–2
O–
3PO
O
O
3PO
CH2
CH2
H O
OH CH2
H O
OH CH2
H
H
H
H
OH
H
OH
H
(F-6-P)
(F-1,6-bP)
FIGURE 7.6 Reaction of phosphorylation of fructose-6-phosphate (F-6-P) catalyzed by PFK.
The Emergence of Temporal Order within a Living Being
175
F-6-P + PFK (ATP)
PFK [(ATP) + (F-6-P)]
F-1,6-bP + PFK [(ATP) + (F-6-P)]
(F-1,6-bP) PFK [(ATP) + (F-6-P)]
(F-1,6-bP) PFK [(ATP) + (F-6-P)]
PFK + 2 (F-1,6-bP) + ADP
F-6-P + PFK (ATP) + F-1,6-bP
PFK + 2 (F-1,6-bP) + ADP
FIGURE 7.7 The autocatalysis of F-1,6-bP.
act
k
PFK
[GLU]
F-6-P ( x)
F-1,6-bP ( y)
1
2
3
[ATP]
ADP
FIGURE 7.8 Scheme of the glycolytic steps involving PFK. [GLU] is the concentration of glucose, which is
assumed to be fixed as that of [ATP]. F-6-P and F-1,6-bP are the x and y variables, respectively, appearing in equation [7.15].
The general differential equations for the mechanism of Figure 7.8 are
dx = v 1− v 2( x, y) = X ( x, y)
dt
[7.15]
dy = v 2( x, y)− v 3( y) = Y ( x, y)
dt
where:
x is F-6-P
y is F-1,6-bP.
The rate of F-6-P formation is v ; (
(
1 v 2 x, y) is the rate of the autocatalytic step; v 3 y) is the rate of
F-1,6-bP consumption. The collection of the signs of the four partial derivatives ( X , X , Y , Y ) is x
y
x
y
referred to as the character of the reaction scheme. Thus, it is easy to determine that X x = − v 2| x, X y = − v 2| y, Yx = v 2| x and Yy = v 2| y − v 3| y. It derives that the determinant of the Jacobian is always positive:
det ( J ) = X Y
2| ( 2|
3| )
x y − X Y
y x = − v x v y − v y + v 2| xv 2| y = v 2| xv 3| y > 0. [7.16]
The trace is
tr ( J ) = X x + Yy = − v 2| x + v 2| y − v 3| y. [7.17]
We have stationary oscillations when tr( J ) = 0, i.e., when Yy = − X x. In other words, it is required that the self-coupling terms are equal in absolute value but with opposite character.
The character of the oscillatory reaction may be qualitatively expressed by a net flux diagram
where the large grey arrows represent the collected sets of pathways that produce and remove the
chemicals (see Figure 7.9).
When the contour conditions impose an inhibitory self-coupling term also to the variable y (i.e.,
Yy = v 2| y − v 3| y < 0), the trace of J becomes negative, and glycolysis exhibits damped oscillations. If, on the other hand, Yy > X x , the trace of J becomes positive and the oscillations become sustained.
TRY EXERCISE 7.1
176
Untangling Complex Systems
dx
X( x,y)
dt
dy
Y( x,y)
dt
FIGURE 7.9 Net diagram flux for a system of two variables giving rise to stationary oscillations. A tradi-
tional arrow, like →, means activation, whereas the symbol ‒‒| means inhibition.
7.3 CELLULAR SIGNALING PROCESSES
All living organisms struggle to survive in an ever-changing environment. They need to collect infor-
mation about the outside world. They regulate their internal machinery for adapting to the external
environment. The fundamental internal control mechanisms are mediated by conformational changes
of proteins receiving either physical (like electromagnetic waves, mechanical forces, and thermal
energy) or chemical stimuli. Such conformational changes of proteins trigger a cascade of enzymatic
reactions, called signal transduction pathway, which ultimately leads to a regulatory or sensing effect.
In fact, molecular signaling processes operate within the cell in metabolic regulation, among cells in
hormonal and neural signaling, and between an organism and the environment in sensory reception.
7.3.1 The simPlesT signal TransducTion sysTem
The most straightforward model of signaling event (Ferrell and Xiong 2001) is shown in Figure 7.10.
In Figure 7.10, the receptive signaling protein, R, is converted to its activated form R* by a stimulus enzyme S. Moreover, R* is turned back to the original state R through an inactivating enzyme P.
For instance, S may be a protein kinase that transfers phosphate groups from ATP to R, producing a
phosphorylated R. The latter is its activated state, R*. The addition of a phosphate group to an amino acid residue usually turns a hydrophobic portion of a protein into a polar and extremely hydrophilic
one. In this way, it can introduce a conformational change in the structure of the protein via interac-
tion with other hydrophobic and hydrophilic residues in the protein. P will be a phosphatase, which
removes the phosphate groups and reverts the signaling protein to its inactivated state, R.
The differential equation describing how the concentration of R* changes over time will be
d R*
= k 1[ S][ R]− k−1[ P] R* [7.18]
dt
where k and
are the kinetic constants of the forward and backward reactions, respectively.
1
k–1
Introducing the mass balance for R ([ ]
*
tot = [ ] + [ *
R
R
R ]) in [7.18] and indicating the ratio [ R ]/[ R] tot
as χ R*, we obtain
dχ
R* = k 1[ S](1− χ ) − −1[ ]
*
k P χ [7.19]
dt
R
R*
+ S
R
R∗
+ P
FIGURE 7.10 The most straightforward transduction mechanism.
The Emergence of Temporal Order within a Living Being
177
k 1[ S]
k
Backward reaction
–1[ P]
Forward reaction
Ra
te
te
Ra
0.0
0.2
0.4
0.6
0.8
1.0
