Ann. Henri Poincaré 4, Suppl. 1 (2003) S459 – S474
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/01S459-16
DOI 10.1007/s00023-003-0936-8
Annales Henri Poincaré
Polymer Transport in Random Potentials
and the Genetic Code: The Waltz of Life
M. Aldana-González, G. Cocho, H. Larralde and G. Martı́nez-Mekler
Abstract. We address the question of why the translation from nucleic acids to protein forming amino acids is carried out by triplets known as codons. We approach
this problem from a dynamical point of view by considering the translocation
properties of primitive molecular machines operating under prebiotic conditions
[1, 2, 3, 4]. Our model captures some basic ribosome-mRNA interaction features.
We consider short chains of charged particles interacting with polymers via electrostatic forces, constrained to move in quasi one dimensional geometries, subject
to external forcing. Our numerical and analytic studies of statistical properties of
random chain/polymer potentials suggest that, under very general conditions, a
dynamics is attained in which the chain moves along the polymer in steps of three
monomers, traversing swiftly two monomers and lingering on the third one as in
a waltz. This behavior is enhanced when we consider present day protein coding
sequences. We also comment on noncoding sequences. We argue that this property
could be one of the underlying causes for the three base codon structure of the
genetic code.
1 Introduction
The genetic code relates nucleic acids from the genetic material to protein constituent amino acids. Key elements of this process are the number m of nitrogenous
basis and the number aa of codified amino acids. Words from one of the alphabets
correspond to elements of the other. In the genetic code these nitrogenous basis
words, known as codons, have a fixed size1 n. One of the main questions regarding the genetic code is to achieve an understanding of why m = 4, aa = 20 and
n = 3. If we take aa = 20 (which is the present number of protein constituent
amino acids) and vary m and n we find several arrangements that allow for the
20 amino acids. For example, in the table below we highlight with bold letters
the simplest choices with an even number of basis (this avoids undecidable situations in a replication process where one base may correspond to more than one
“complementary” base). The first column in the table is the number of elements
in the codon, the second is the size of the nucleic acid alphabet, the third shows
the number of nitrogenous base combinations (amino acid words) and the third
indicates whether or not there is complementarity.
1 Though it has been pointed out [5] that more efficient codes can be obtained by allowing the
length of codons to vary.
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M. Aldana-González et al.
n
1
2
2
2
3
3
3
4
4
m
20
4
5
6
2
3
4
2
3
aa
20
16
25
36
8
27
64
16
81
Ann. Henri Poincaré
complementarity
yes
yes
no
yes
yes
no
yes
yes
no
Rows 1, 4 and 7 are possible choices. It would be desirable to have some
“first principle” argument for choosing a value of n independently from m, or
viceversa. This is the main motivation for this work. Here we review a proposition
put forth by us [1, 2, 3, 4] where n is associated to the value of 3 based on the
dynamical properties of primitive synthesis molecular machines which would be
operating under pre or protobiotic conditions. We will only highlight the main
lines of argument, detailed descriptions can be found in the above references. The
general setup relies on the following considerations:
• Protein synthesis takes place at the ribosome which is a macromolecular
cell structure where a polymer, the messenger ribonucleic acid (mRNA),
is translocated through the ribosome along a tube-like geometry. The path
along which the mRNA slides is mainly conformed by ribosomal ribonucleic
acid (rRNA). The ribosome acts as a complex molecular machine that moves
along the mRNA.
• A primitive synthesis and replication molecular machinery may have consisted of polymers (mRNA antecedents) moving in quasi 1-dimensional channels (ribosomal tube antecedents).
• The relative rRNA/mRNA motion corresponds to the relative movement between free polymers and polymers fixed to the channel. Both polymers would
be constituted by various electrically charged monomers. Motion would be
induced by external electromagnetic fields.2
• The transport properties of such primitive molecular machines would condition chemical reactions and coding processes.
Concerning the last two points we consider the generic situation of polymers with randomly charged monomers. We first look at their mutual interaction
potentials and then focus on the statistical properties of the distance between
2 In [1, 2, 3] we argue in favor of the plausibility of an outer space origin of life scenario [6]
where interstellar dust or meteorite surfaces would provide the quasi 1 dimensional templates
and UV radiation would charge the monomers
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Polymer Transport in Random Potentials and the Genetic Code
S461
neighboring potential minima. We then show how these static properties influence
the statistical characterization of dynamical quantities via an equation of motion.
Our central result is that under the above conditions, a dynamics is attained
in which the polymer moves along the channel in effective “steps” whose mean
length is three monomers. We suggest that this dynamical feature may be an
important underlying cause of the three base codon structure of the genetic code.
2 The model
We propose a model consisting of a chain of M monomers, with charges {pi },
interacting with a very long polymer composed of N monomers with charges {qj },
as shown in Fig. 1. The charges we consider are not necessarily Coulomb charges,
both {pi } and {qj } could be dipolar moments, induced polarizabilities, or similar
quantities resulting from electrostatic interactions between chain monomers and
polymer monomers with potentials of the form 1/rα , where α characterizes the
“charge” type (α = 1 corresponds to an ion-ion interaction, α = 2 represents an
ion-dipole interaction, and so forth). We do assume that all the charges are of the
same nature. We consider the charges {pi } and {qj } to be discrete independent
random variables, acquiring one of m different values with the same probability.
All monomers have the same length L and their charge is taken to be uniformly
distributed along their length. The interaction potential Vij (x) between the ithmonomer in the chain and the jth-monomer in the polymer is given by
x+i
x+i−1
dx dx
j
Vij (x) = Kpi qj
j−1
α/2
[(x − x )2 + σ 2 ]
(1)
where K is a constant whose value depends on the unit system used to measure
the physical quantities. We have taken L = 1 in the above expression. The overall
interaction potential Vσα (x) between the whole chain and the entire polymer is
given by the superposition of the individual potentials Vij (x):
Vσα (x) =
M N
Vij (x)
(2)
i=1 j=1
3 Statistical Properties of Random Potentials:
We start with the simplest M = 1 case, in which the chain consists of just one
monomer. We will refer to this as the particle-polymer case. Fig. 2 shows three
graphs of the potential Vσα (x) for σ = 0.5 and different values of the parameter α. Charges {qj } are uniformly distributed amongst m = 6 different values
{±1, ±2, ±3}. Notice that the distribution of maxima and minima along the potential does not change significantly by varying the value of the parameter α, in
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M. Aldana-González et al.
Ann. Henri Poincaré
.
...
p1
x
pM
Fex
σ
q1 q2
q3
...
qN
.
Figure 1: Chain composed of M monomers interacting with a polymer, N monomers in length.
Charges {pi }, {qj } are independent random variables. The chain is at a fixed perpendicular
distance σ from the polymer, but can move along the polymer. x is the relative polymer/chain
horizontal position.
the sense that all the maxima and minima remain essentially at the same positions. As α takes on larger values, the potential approaches a step-like function.
A similar behavior is encountered if instead of varying the interaction range we
decrease the distance from the particle to the polymer and keep α constant.
We are mainly concerned in the statistical behavior of the distance d between
¯ the most probable
consecutive minima, since by looking at the average value d,
∗
value d and the probability distribution function of d, we will obtain important
dynamical information of the system. If an external force is acting on the particle
(or the chain), forcing it to move in one direction along the polymer, the particle
will spend more time in the energy minima than in the maxima. Such a movement
may be interpreted by considering the particle as “jumping” from one minimum
to the next (see Fig. 3).
Using the step-like limit, we have derived[1] that for a long polymer (in the
large N limit) d¯ is given by:
d¯ =
6m
2m − 1
(3)
where we recall that m is the number of different monomer charges or monomer
types if we use the charge as a monomer label. The mean distance d¯ is hence always
between 3 and 4, approaching 3 asymptotically as m → ∞.
For the probability distribution function Pm (d), of two consecutive minima
being separated by a distance d when there are m different types of monomers we
have closed expressions [4] valid for the step-like limit:
2k + 1
6
Pm (d) =
Nm (d − k)
2m − 1
mk+d+2
d−2
k=0
(4)
Vol. 4, 2003
Polymer Transport in Random Potentials and the Genetic Code
25
15
5
−5
−15
−25
0
20
40
S463
60
80
100
60
80
100
60
80
100
(a)
10
5
Vσα(x)
0
−5
−10
0
20
40
(b)
5
3
1
−1
−3
−5
0
20
40
(c)
x
Figure 2: Particle-polymer interaction potential for σ = 0.2 and different values of α. (a)
α = 1; (b) α = 3; (c) α = 5. Charges along the polymer, selected at random, are the same in the
three graphs shown. Note that as α increases, the potential becomes a step-like function, but the
positions of the maxima and minima do not change. There are 33 potential minima distributed
along 100 monomers, giving a mean distance between consecutive minima d¯ = 100/33 = 3.03.
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M. Aldana-González et al.
Ann. Henri Poincaré
V(x)
111
000
000
111
000
111
000
111
000
111
x
Figure 3: In a one dimensional potential, the dynamics is determined by the distribution of
maxima and minima.
if d is integer: d = 2, 3, 4, . . ., and
d −2
12 2k + 1
Pm (d) =
Nm (d − k)
2m − 1
mk+d +3
(5)
k=0
if d is half-integer: d = 5/2, d = 7/2, d = 9/2, . . ., where d = Int[d]. In the above
expressions, d is measured as the distance between the midpoints of the adjacent
minima and Nm (d) is a polynomial whose degree and coefficients depend on m.
For m = 2, m = 3 and m = 4 the polynomials
are given by N2 (d) = 1,N3 (d) =
1 + 2d + 2d2 and N4 (d) = −2 + 1/8 30d + 73d2 + 22d3 + 3d4 .
For the case of m = ∞ we have also derived a closed expression [4], which
has a much simpler form:
P∞ (d) = 3
2d 2
d +d−2
(d + 3)!
(6)
The preceding distributions, which are plotted in Fig. 4, show that the most
probable distance d∗ between consecutive potential minima is d∗ = 2, except for
the case m = 2 in which d∗ = 3. Hence, according to the transport mechanism
suggested in Fig. 3, whenever there are more than two different types of monomers,
the particle will move along the polymer in “jumps” whose mean length is close
to three, but whose most probable length is actually two. The difference between
the mean distance d¯ and the most probable distance d∗ is due to the presence of
“tails” in the probability function Pm (d). Namely, to the fact that Pm (d) has non
zero values even for large d. However, these “tails” can be considerably shrunk [4]
when the chain is made up of more than one particle (M > 1).
We now consider the case M > 1. This situation is closer to the present day
ribosome based dynamics since at each given time, the ribosome interacts with the
mRNA at several points, giving rise to a multiple interaction. Electron microscopy
has revealed that the mRNA thread passes across the ribosome through a tunnel
about 10nm long and 2nm in diameter [8]. Since nucleotide length is about 0.5nm,
Vol. 4, 2003
Polymer Transport in Random Potentials and the Genetic Code
0.20
0.15
0.10
0.05
0.00
S465
m=2
0
2
4
6
8
10
12
14
0
2
4
6
8
10
12
14
0
2
4
6
8
10
12
14
0
2
4
8
10
12
14
0.30
m=3
P(d) 0.20
0.10
0.00
0.30
m=4
0.20
0.10
0.00
0.40
0.30
0.20
0.10
0.00
m= ∞
6
d
Figure 4: Probability function Pm (d) of two consecutive minima being separated by a distance
d, for different values m of monomer types.
around 20 nucleotides may be in a position to interact simultaneously with the
ribosome. Taking the above into account, we will work with a small chain, assuming
M = 10 as a reasonable length.
Numerical simulations of Pm (d) show that it has a strongly fluctuating behavior. Its shape depends crucially on the particular realization of monomer charges
in the chain. For example, Fig. 5a shows Pm (d) for a polymer 500 monomers in
length and a random realization of charges in a 10-monomer chain. Charge values
were {±1, ±2}. Fig. 5b shows a similar graph for a different realization of charges
along the chain. Polymer charges were kept the same in both cases. Note that the
first graph presents a very sharp maximum at d∗ = 2, whereas the second one
does so at d∗ = 3. When we consider 4500 charge realizations for a 10-particle
chain interacting with a 500 monomer polymer (typical length of sequences coding
functional proteins [8]) we obtain the following table for the occurrence of either
d∗ = 2 or d∗ = 3:
d∗
d =3
d∗ = 2
∗
Fraction of Occurrences
0.53
< 0.33
Table I
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0.8
0.6
P(d) 0.4
0.2
0
0
1
2
3
4
5
(a)
6
5
(b)
6
7
8
9
10
d
0.8
0.6
P(d) 0.4
0.2
0
0
1
2
3
4
7
8
9
10
d
Figure 5: Probability functions P4 (d) corresponding to two different realizations of monomer
charges for a 10-monomer chain interacting with a 500 monomer polymer. Mean distance is
d¯ 2.36 for (a) and d¯ 2.98 for (b).
The above result shows that when a multiple interaction between the polymer and the chain is considered, the probability of having a random interaction
potential whose consecutive minima are more often separated by three monomers,
is dominant.
4 Statistical Properties of Genetic Sequence “Potentials”
So far we have studied generic behaviors in the sense that they refer to almost
all possible electrostatic interactions and hold for randomly associated monomer
charges. If we now assign the charges {qj } along the polymer in correspondence
with the genetic sequence of an organism via a lookup table, we may test the
model’s response to the genetic information accumulated along evolution. Since
genetic sequences are made out of four different bases (A, U, C and G) we consider
four different possible values for the charges {qj }, i.e. m = 4. An arbitrary look
up table is the following:
base
A
U
C
G
charge value
−2
−1
+1
+2
(7)
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Polymer Transport in Random Potentials and the Genetic Code
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0.3
P(d) 0.2
0.1
0.0
0
1
2
3
4
5
(a)
6
5
(b)
6
7
8
9
10
d
0.3
P(d) 0.2
0.1
0.0
0
1
2
3
4
7
8
9
10
d
Figure 6: Probability function P4 (d) computed by assigning the charges in the polymer in
correspondence with genetic sequences of the Drosophila malanogaster genome: (a) proteincoding, d¯ = 3.15, (b) intergenic, d¯ = 3.43.
Figure Fig. 6a shows the probability distribution P4 (d) computed numerically
by using a Drosophila melanogaster protein-coding sequence, 45500 bases in length
(several genes were concatenated to construct this sequence) considering the M =
1, particle-polymer situation. The mean distance between consecutive potential
minima for this sequence is d¯ 3.15 while the most probable distance is d∗ = 3.
Therefore, even for particle-polymer interactions, in the “real sequence case” not
only is the mean distance d¯ very close to 3, but also the most probable distance
d∗ turns out to be 3. A comparison of the m = 4 case of Fig. 4 with Fig. 6a, shows
that for protein-coding genetic sequences, the potential minima along the polymer
are more often separated by three monomers than in the random case, the value
of P4 (d) increases when d = 3 and decreases at d = 2.
The above behavior does not occur when non-coding sequences of the genome
are used for the monomer charge assignment along the polymer. Fig. 6b shows
P4 (d) for an intergenic sequence of the Drosophila’s genome. The length of the
sequence is again 45500 bases. In this case P4 (d) looks much more like the one
obtained for the random case.
It is important to remark that the behavior of P4 (d) exhibited in Fig. 6a for
real protein-coding sequences does not depend on the particular correspondence
between bases and charge values being used, as long as they are of similar order of
magnitude and they can allow for an order relation.3 Furthermore, the probability
3 The
four bases A, T, C and G have charges of the same order of magnitude [7].
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Ann. Henri Poincaré
function P4 (d) is invariant under base permutations [4], this also holds for noncoding sequences.
In reference [4] we looked at several organisms of a variety of types and found
that the maximum at d∗ = 3 of P4 (d) is a general characteristic associated with
the protein-coding sequences of living organisms.4 The fact that the above characteristic is absent in non-coding genetic sequences may be interpreted in evolutionary terms. Genetic sequences directly involved in the protein-translation processes
were selected (among other things) as to bring the distance between consecutive
potential minima closer to 3, both in mean and frequency of occurrence. This process may have been enhanced by multiple chain-polymer interactions which, as
pointed out in connection with Table I, favor a value of d∗ = 3.
5 Dynamics
Ribosome translocation is an intricate process dependent on many physical and
chemical factors [8]. A detailed modeling of this process is practically unmanageable. However, as with other complex phenomena, it is fruitful to focus on
specific matters relevant at a given level of description. For example, the recent
experimental evidence suggesting a ratchet-like behavior [9] is indicative of the
appropriateness of an analysis within the perspective of the current developments
on the study of ratchet dynamics [10]. In this work, we have chosen to look into
the behavior of oversimplified molecular models which might capture some essential dynamical features of the system and shed some light on how this mechanism
could have arisen.
Our modeling of the dynamics of the system is governed by the application
of an external force Fex to the chain in the horizontal direction, i.e. parallel to
the polymer. By this means the chain will be forced to move along the polymer.
In principle, the force Fex may be time dependent, but we will restrict ourselves
to a constant term. This force might come from a chemical pump or from any
other electromagnetic force which may have been present in prebiotic conditions. It
should be strong enough as to drive the chain along the polymer (which is assumed
to be fixed, N → ∞ limit), without getting trapped in some of the minima of the
polymer-chain interaction potential V (x), and weak enough in order to “detect”
the characteristics of the potential. Therefore, we assume that |Fex | is slightly
larger than the maximum value of |∂V (x)/∂x|.
Our analysis relies on Newton’s equation of motion in a high friction regime,
where inertial effects can be neglected. This regime actually exists in biological
molecular ratchets similar to the one we are considering [10]. Under such conditions, the equation of motion acquires the form
γ
∂V α (x)
dx
=− σ
+ Fex
dt
∂x
(8)
4 This property is a consequence of the order in which the bases or codons appear along the
sequence and is not related to the relative weight (fraction) of their occurrence[4].
Vol. 4, 2003
Polymer Transport in Random Potentials and the Genetic Code
S469
150
100
v(x)
50
0
40
50
60
(a)
70
40
50
60
(b)
70
80
x
40
t(x) 20
0
80
x
Figure 7: (a) Particle velocity along the polymer as a function of the position x. (b) Local
transit time t as a function of the position x.
where γ is the friction coefficient. In what follows, we will set γ = 1, which is equivalent to rescaling the time units. The above, though a deterministic equation, gives
rise to a random dynamics inherited from the randomness of the interaction potential Vσα (x). For the particle polymer case we show in Fig. 7a a typical realization
of the velocity of the particle v(x) as a function of its position x along the polymer
determined from equation (8), with α = 2 and σ = 0.5, and monomer charge values of {±1, ±2} (case m = 4). Fig. 7b shows the local-transit times of the particle
along a short segment of the polymer (40 monomers in length) computed by counting how many time steps the particle spent in every spatial interval ∆x = 0.25. It
is clear from this figure that the particle spends more time in certain regions than
in others, the former being more or less regularly spaced along the polymer.
Spatial regularities in the dynamics of the system appear when we consider
the particle velocity Fourier power spectrum |v̂(k)|2 , which is shown in Fig. 8 for
different realizations of monomer charges in the polymer. The parameter values in
Fig. 8a and Fig. 8b are {σ = 0.5, α = 1} and {σ = 0.5, α = 4} respectively. Charge
values are {±1, ±2}. Notice the dominant frequency k ∗ in the power spectrum of
the velocity (the highest peak), whose corresponding spatial periodicity is λ∗ =
2π/k ∗ ∼ 3.3.
If we now consider the power spectrum of the velocity of the particle throughout the polymer, using as in the previous section protein-coding sequences, we find
a behavior as shown in Fig. 9 for the tuberculosis bacillus. This graph was generated using coding sequences of 10000 monomers. The dominant sharp peaks are
much higher now than for the random sequences and closer to 3. In reference [4]
we show that this behavior is recurrent for different types of organisms.
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1
k * = 1.85
0.8
|v(k)| 2
λ* = 3.29
0.6
0.4
0.2
0
0
2
4
6
k
(a)
1.5
k * = 1.90
|v(k)| 2
λ* = 3.30
1
0.5
0
0
2
4
6
k
(b)
Figure 8: Power spectrum of the velocity of the particle along the polymer for two random
monomer charge realizations, λ∗ = 2π/k ∗ .
Tuberculosis Bacillus
0.03
∗
λ =2.99
0.02
|v(k)|
2
0.01
0.00
0
1
2
3
k
4
5
6
Figure 9: Power spectrum of the velocity of the particle using protein-coding sequences of
tuberculosis bacillus to assign the polymer charges. The sharp peak corresponds to a spatial
λ∗ = 3. The peak around k = 4 is the second harmonic.
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Caenorhabditis elegans
8
6
|v(k)| 2 4
2
0
1
2
1
2
3
4
Saccharomyces cerevisiae
5
6
5
6
8
6
|v(k)| 2 4
2
0
3
4
k
Figure 10: Power spectrum of the velocity of the particle using non-coding sequences of S.
cerevisiade (yeast) and C. elegans (worm). In this case the spatial periodicity is weaker (or
absent) than in Fig. 16. It seems that the non-coding sequences of real organisms have a random
like structure.
In Fig. 10 we show the velocity power spectra for intergenic non-coding sequences which turn out to be similar to the ones obtained in the random case. In
this sense, intergenic regions again appear to have a random structure.
The most interesting dynamics occurs when an extended chain is interacting
with the polymer. Under this situation, a multiple interaction prevails. At every
moment there are several contact points between the chain and the polymer. As
we have already pointed out, the multiple interaction between the chain and the
polymer gives rise to a widely fluctuating probability function Pm (d). The same
occurs with the power spectrum of the velocity of the chain along the polymer.
In Fig. 11 we show the power spectra of the velocity of the chain along the
polymer for two different random realizations of monomer charges in the chain. The
charges in the polymer were the same in both cases. These graphs were constructed
with a polymer 500 monomers in length and a 10-monomer chain. The parameter
values used were σ = 0.5 and α = 1 with charge values {±1, ±2}. The figure shows
considerable structure with dominant frequencies, even though the charges in both
the polymer and the chain were assigned at random. In Fig. 11a this frequency
corresponds to a a spatial periodicity λ∗ 2, whereas in Fig. 11b to λ∗ 3. The
difference in these values may be related to the fact that for case (a) P4 (d) has a
maximum at d∗ = 2, while for case (b) this occurs at d∗ = 3. Further numerical
exploration shows that whenever the probability function has a sharp maximum
at a distance d∗ , the power spectrum of the velocity also presents a sharp peak
corresponding to a spatial periodicity λ∗ = d∗ . This observation, together with
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M. Aldana-González et al.
Ann. Henri Poincaré
40
k *= 3.14
30
|v(k)| 2
λ* = 2.01
20
10
0
0
2
4
(a)
6
8
k
60
k *= 2.18
|v(k)| 2
λ* = 2.88
40
20
0
0
2
4
(b)
6
8
k
Figure 11: Power spectrum of the velocity of a 10-monomer chain along a polymer 500 monomers
in length for two realizations of monomer charges in the chain.
the dominance of probability density functions P4 (d) with d∗ = 3 for multiple
interactions, implies that when we assign charges at random to the monomers
in the polymer and in the chain, the most general outcome will be a dynamics
registering three monomer lengths.
6 Summary and discussion
The results presented throughout this work suggest a possible scenario for the
origin of the three base codon structure of the genetic code. In this scenario, the
relative movement between a long and a short polymer in a quasi one dimensional
geometry (one of them fixed to it), with a random interaction potential, subject to
external forcing, exhibits a regular dynamics with a “preference” for a movement
in steps of three bases. By “steps” we mean a slowing down of the relative velocity
nearly every three monomers (see Fig. 7b).
The preceding property is quite robust inasmuch as it hardly depends on the
particular kind of electrostatic interaction between the polymer and the chain.
The spatial distribution of interaction potential minima does not depend on the
particular values of the monomer charges. Furthermore, for the polymer-particle
case, the probability function Pm (d), which gives the probability of two consecutive
minima being separated by a distance d, only depends on m, namely, the number
of different types of monomers. As m increases, the mean distance d¯ between
consecutive potential minima approaches three. Nevertheless, the most probable
distance is d∗ = 2 (for m > 2).
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Polymer Transport in Random Potentials and the Genetic Code
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Considerable changes take place when the charges along the polymer are assigned in correspondence with protein-coding genetic sequences of real organisms.
In this case, not only is the mean distance d¯ between neighboring potential minima nearly three, but also the most probable one, d∗ , happens to be three. This is
a remarkable property of protein-coding sequences, perhaps acquired throughout
evolution. The fact that this “refinement” is absent in non-coding sequences of real
organisms, strongly suggests that it is a consequence of the dynamical processes
involved in the protein synthesis mechanisms.
This interpretation is supported by the results obtained when the dynamics
of the particle moving along the polymer is considered. In the random sequence
case, there are dominant frequencies in the power spectrum of the particle velocity
related with the spatial regularities of the interaction potential. Moreover, in the
protein-coding sequence case, the power spectrum of the velocity shows a very well
defined peak corresponding almost exactly to a spatial distance λ∗ = 3. Again,
this behavior does not occur for non-coding sequences of real organisms, which are
not involved in the translation processes.
A richer dynamics emerges when the chain is composed of several monomers. In this, more realistic, multiple-interaction case, the probability function
Pm (d) presents very wide fluctuations, depending on the particular assignment of
monomer charges in the chain. Nevertheless, the most encountered configurations
are those for which the probability function has its highest value at d∗ = 3. For
these configurations, the power spectrum of the chain velocity along the polymer
is indicative of a well defined spatial periodicity at λ∗ 3.
Our results suggest an origin of life scenario in which primordial molecular
machines of chains moving along polymers in quasi one-dimensional geometries,
that eventually led to the protein synthesis processes, were biased towards a dynamics favoring the motion in “steps” or “jumps” of three monomers. The higher
likelihood of these primitive“ribosomes” may have led to the present ribosomal
dynamics where mRNA moves along rRNA in a channel conformed by the ribosome. Thus, dynamics may have acted as one of the evolutionary filters favoring
the three base codon composition of the genetic code.
Acknowledgments
This work was sponsored by the DGAPA-UNAM project IN103300, the MRSEC
Program of the National Science Foundation (NSF) under Award Number DMR
9808595 and by the NSF Program DMR 0094569. M. Aldana also acknowledges
CONACyT-México a posdoctoral grant.
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M. Aldana-González
The James Franck Institute
The University of Chicago
5640 South Ellis Avenue
Chicago, Il, 60637, USA
G. Cocho
Instituto de Fı́sica
UNAM. Apdo. Postal 20-364
01000 Mexico D.F., Mexico
H. Larralde
Centro de Ciencias Fı́sicas
UNAM. Apdo. Postal 48-3
62251 Cuernavaca, Morelos, Mexico
G. Martı́nez-Mekler5
Facultad de Ciencias
Universidad Autónoma
del Estado de Morelos
Av. Universidad 1001
Cuernavaca, Morelos, Mexico
5 Permanent address: Centro de Ciencias Fı́sicas, UNAM, Apdo. Postal 48-3, 62251 Cuernavaca, Morelos, Mexico