Holographic bound and protein linguistics
Holographic bound and protein linguistics
Dirson Jian Li∗ and Shengli Zhang
Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, PR China
(Dated: November 16, 2018)
The holographic bound in physics constrains the complexity of life. The finite storage capability of
information in the observable universe requires the protein linguistics in the evolution of life. We find
that the evolution of genetic code determines the variance of amino acid frequencies and genomic
GC content among species. The elegant linguistic mechanism is confirmed by the experimental
observations based on all known entire proteomes.
PACS numbers: 87.10.+e, 87.15.Aa, 87.14.-g, 04.70.Dy
The phenomenon of life is governed by the general prin-
ciples in physics, so the progress in understanding the
physical world may provide new insight into the origin
and evolution of life. The analogy between biology and
linguistics at the level of sequences hints that the bio-
information is processed by underlying linguistic rules.
Several attempts have been made to combine linguistic
theory with biology [1]. But the existence of linguistics
in the biomacromolecular sequences needs a physical ex-
planation. The holographic bound, intimately related to
the holographic principle, came from the deep insights
of Bekenstein and Hawking in 70’s [2][3]. Its validity
is insured by the second law of thermodynamics. In-
terestingly, the problem on the existence of linguistics
in the biomacromolecular sequences can be explained by
the holographic bound. In the past decades, the biology
has been changed greatly. Wada advocated “... to deter-
mine the ‘first principles’ of bio-sciences and link them
with the first principles of non-bio-sciences in order to
understand the complex systems.” and Gilbert also em-
phasized the importance of the theoretical methods in
biology [4]. Nowadays, the intimacy between biology and
physics is unprecedented. Considering the significant role
of information either in physics or in biology [5][6], the
gap between physics and biology may be bridged from
the viewpoint of information.
In the post-genomic era, the number of entire pro-
teomes increases rapidly. We can take all known entire
proteomes as samples to study the global properties of life
on our planet. The variance of GC content [7] is a global
property, which varies greatly among species. We also
found another global property of the evolution of amino
acid frequencies though they vary slightly. The mecha-
nism of the variance of genomic GC content and amino
acid frequencies was a long-standing and far-reaching
problem [8]. The genetic code evolved in the context of
four-letter alphabet when the 20 amino acids joined pro-
tein sequences chronologically [9]. The nature of prime
biased AT/GC pressure and the reason for the correla-
tion of GC content between total genomic DNA and the
1st, 2nd and 3rd codon positions were unknown. The
profound mechanism behind the variance of amino acid
frequencies has not been studied; it is worse that the
amino acid frequencies are routinely assumed to be con-
stant. All of these basic problems in biology are solved in
our theoretical framework based on the formal linguistics
and the evolution of genetic code.
In this paper, firstly we explain the existence of pro-
tein linguistics and the limited complexity of life in the
universe in terms of the holographic bound. Secondly, a
linguistic model is proposed to reveal the mechanism of
the evolution of amino acid frequencies and genomic GC
content as well as the protein length distribution. The ex-
cellent fit between our simulations and the experimental
observations strongly supports the linguistic mechanism,
where the experimental observations are based on the
data of 106 entire proteomes (85 eubacteria, 12 archae-
bacteria, 7 eukaryotes and 2 viruses) in database PEP
[10] and the data of GC content in database Genome
Properties system [11]. The “information” is the thread
of the paper, which connects traditionally irrelative prob-
lems in physics and biology with each other.
According to the holographic bound, which states that
the information storage capacity of a spatially finite sys-
tem is limited by a quarter of its boundary area measured
in Plank area unless the second law of thermodynamics
is untrue, the entropy S in a volume of radius R satisfies
S 6 Smax ≈ (
)2, (1)
where lp is the Plank length. There is much astronom-
ical evidence that our universe may be headed for an
infinite deSitter space. The holographic bound can be
applied to the observable universe with finite event hori-
zon. Therefore we can estimate the upper limit of the
information storage capability of the observable universe
as Iuniv ≈ 10
122 bits [3]. In the point of view in physics,
the entropy in our universe is given primarily by the num-
ber of black body cosmic background photons, ∼ 1090,
which is definitely less than Iuniv.
However, the upper limit of information Iuniv is not
so large when considering the information of a system of
life. Firstly, let us give a reasonable definition of a “liv-
ing” system from the viewpoint of information. Many
restless functional proteins, composed of 20 amino acids
(a.a.), distinguish the life from the lifeless matter. Thus,
http://arxiv.org/abs/0704.1169v1
a general living system L(n) is defined as a set of all pos-
sible proteins with length no more than n a.a., each of
which is in either the folded state or the unfolded state.
The maximum length n indicates the complexity of the
system. Then, let’s calculate the information of the sys-
tem. The number of states of L(n) is Ω(n) =
k=2 2
hence its information is
I(n) = log2Ω(n) ≈ 20
n bits. (2)
The exquisite single-chain structure of proteins can pro-
vide much more information storage capacity than life-
less matter. The upper limit of information Iuniv forbids
L(n), n > n0 to exist in our universe, where n0 = 94 a.a.
such that I(n0) ∼ Iuniv. Interestingly, the most frequent
protein length for the life on our planet is about n0.
The actual system of life on the planet is not one of
L(n), because the average protein length for different
species, ranging from 250 a.a. to 500 a.a., are greater
than n0. Let the actual living system Learth be the
set of all possible proteins on the planet, and suppose
n∗ (> n0) be the maximum protein length. Learth must
be a proper subset of L(n∗) because the information of
Learth is bounded by Iuniv . According to the linguis-
tic theory, Learth is a language over the alphabet of 20
amino acids. So we have demonstrated the existence of
protein linguistics in terms of the holographic bound that
can be derived from the second law of thermodynamics.
In early evolution when delivering the genetic informa-
tion from the RNA world to the DNA-protein world, the
holographic bound required the grammars to allow a very
small part of protein sequences and to forbid all the oth-
ers. There is a easy way to implement the forbiddance:
the observable universe can not accommodate all the pro-
teins in L(n∗). In fact, our conclusion is based on the
general principle, freedom of the subtleties of the hierar-
If L(n) have an additional property, e.g., chirality,
Lr(n) (Ll(n)) become the set of proteins composed of
right-handed (left-handed) amino acids. Let C(n) be the
set of all possible chiral L(n). Only up to n = 2 a.a.,
the information of C(2), i.e., IC(2) = Ω(2) ≈ 10120 bits
is near Iuniv . Chirality might have brought too much
redundant information; broken symmetry was the solu-
tion. So entropy bound is a strong law which constrains
the forms of possible life in general. Let us imagine the
most complex creature with the same height of us be the
one whose genetic information is stored in Plank scale.
The information stored in its body can be estimated as
1/l3p ∼ 10
105 bits, which violates the holographic bound.
So such complex creature can not exist.
So far, we are aware that the linguistics must play a
significant role in generating proteins in the primordial
time. In the following, a linguistic model is proposed to
simulate the variance of the amino acid frequencies and
the genomic GC contents as well as the protein length
distributions.
   G   A   D   V   P   S   E   L   T   R   Q   I   N   H   K   C   F   Y   M   W    
   G   A   D   V   P   S   E   L   T   R   Q   I   N   H   K   C   F   Y   M   W    
(for each a.a.)
     1...106   
(for each a.a.)
     1...30
FIG. 1: Evolution of amino acid frequencies. The 20
amino acids are aligned chronologically. The variance for each
amino acid in simulation fits the experimental observation.
(a) Experimental observation base on the data of 106 species.
For each amino acid, the 106 species are aligned from left to
right by R10/10. (b) Simulation by the linguistic model. The
30 simulated proteomes are aligned by t, which increases from
0.02 to 0.40 by equal steps. (The simulations in Fig. 1, Fig.
2 and Fig. 4 are obtains together by the linguistic model.)
The model consists of three parts: (i) generate protein
sequence by tree adjoining grammar [12]; (ii) set amino
acid for the leaves of grammars in (i) according to the tree
of genetic code multiplicity (can be obtained from sym-
metry analysis, see [13]) with consideration of the amino
acid chronology [14]; and (iii) translate the protein se-
quences to the DNA sequences according to genetic code
chronology [15]. The evolution of genetic code is the core
of the model. There is a variant t in the model, which
represents the time in evolution. A proteome for a species
is defined as many a protein generated by the model with
fixed t, so t also identifies species in the model. Thus, the
amino acid frequencies and the average protein length
for a species can be calculated. The evolutionary trends
of the amino acid frequencies can be determined when
proteomes are generated at different time t. We can also
simulate the evolution of genomic GC content after trans-
lating the protein sequences to DNA sequences.
The evolution of amino acid frequencies can be ex-
plained by the model. According to the consensus
chronology of amino acids to recruit into the genetic code
from the earliest to the latest [14]: G, A, D, V, P, S, E,
L, T, R, Q, I, N, H, K, C, F, Y, M, W, we sort the
106 species by the ratio R10/10 of average frequency for
10 later amino acids to average frequency for 10 earlier
amino acids. Then we obtain the evolutionary trends of
amino acid frequencies: the frequencies of G, A, D, V,
P, T, R, H, W decrease, while the frequencies of S, E, I,
N, K, F, Y increase and the frequencies of L, Q, C, M
do not vary obviously (Fig. 1a). The variance of amino
0.2 0.4 0.6 0.8
Genomic GC content
0.2 0.4 0.6 0.8
Genomic GC content
convex 
FIG. 2: Evolution of genomic GC content. (a) Relation-
ship between genomic GC content and R10/10 for the species
in database PEP (dots) and its simulation by the linguistic
model (solid line, also decreasing). (b) Simulation of the cor-
relation of the GC content between total genomic DNA and
the 1st, 2nd, and 3rd codon positions, which agrees with the
experimental observation in detail (see fig. 5 in [17], fig. 2 in
[7] and fig. 9-1 in [5]).
acid frequencies are amazingly monotonic by and large.
Therefore, it is reasonable to assume that a mechanism
underlies the evolution of amino acid frequencies.
The simulation of our linguistic model [Fig. 1b] agrees
with the data of 106 species [Fig. 1a] not only in the evo-
lutionary trends but also in the variance magnitudes for
most of the amino acids. Note that no parameter is added
on purpose in the model to alter the trend for a certain
amino acid. The evolution of amino acid frequency are
sensitive to the amino acid multiplicities [13], any disobe-
dience of which would spoil the results. Therefore, it is
the evolution of genetic code that determines the evolu-
tion of amino acid frequencies. An important property of
the model is that the parameters of amino acid frequen-
cies are constant, which indicates that the variance of
amino acid frequencies developed during a short period.
It agrees that the genetic code had accomplished quickly.
Recently, Jordan et al observed the contemporary amino
acid gain and loss, about which there were different ex-
planations [16]. We believe that the evolution of genetic
code drives the amino acid gain and loss.
The genomic GC content decreases linearly with R10/10
for the species in database PEP. The simulation of our
model agrees qualitatively with this experimental obser-
vation [Fig. 2a]. In our model, the evolution of amino
acid frequency and the evolution of genomic GC content
are driven by a common variant t. A protein sequence
generated at later time t corresponds to the DNA se-
quence translated using the later codons [15], which re-
sults in the relationship between genomic GC content
250 350 450 550
Average protein length
Archaebateria
Eubacteria 
Eukaryote 
Mycoplasma 
Virus 
FIG. 3: Rainbow distribution. Relationship between the
average protein length and the highest frequency of the dis-
crete fourier transformation of protein length distribution (the
cutoff is 3000) for each of the 106 species. The distribution
of the species from three domains likes a rainbow. Even for
the group of closely related species such as mycoplasmas (be-
longing to eubacteria), their distribution also form an “arch”
of the rainbow.
and R10/10.
And the simulation of the correlation of GC content
between total genomic DNA and the first, second, and
third codon positions [Fig. 2b] also agrees with the re-
sults based on the data of completed genomes [7][5][17],
where the correlation slope of the third codon position
is much greater than that of the first and the second
positions. There is a characteristic convex in the mid-
dle of the line of the simulated GC content for the first
codon position, which agrees dramatically with the ex-
perimental observations [7][5][17]. In the table of codon
chronology [15], G and C (A and U) occupy all the third
positions of earliest (latest) codons for 20 amino acids,
while the bases appear about equally for the first and
second positions. Therefore, the correlation slope for the
first and second positions vary slightly while the slope
for the third position varies greatly. And the lower limit
l ∼ 0.3 and upper limit u ∼ 0.7 of the GC content among
species can be explained similarly; l + u = 1 is required
in theory.
The linguistic mechanism can also be supported by the
distribution of protein length. When observing the dis-
tribution of species in the space of average protein length
and the highest frequency of discrete fourier transfor-
mation of protein length distribution, we unexpectedly
found that the species for the three domains gathered in
three parallel arches respectively, which likes a rainbow
[Fig 3]. This indicates that the fluctuations in protein
length distributions can not be products of a stochastic
process. The characters of the protein length distribu-
tion (bell-shape profile, periodic-like fluctuations) have
250 350 450 550
Average Protein Length
10 20
Average mini−protein length
Archaebateria
Eubacteria 
Eukaryote 
Virus 
 increasing  t
FIG. 4: The evolutionary flow. Relationship between av-
erage protein lengths and RHQW/GV for the 106 species. The
species of three domains (Archaebacteria: blue square, Eu-
bacteria: dot, Eukaryotes: red circle) gather together in re-
spective regions and all the species form an evolutionary flow.
The proteome size is represented proportionally by the tail
under each species (big: red, middle: green and small: blue);
species with big genome sizes locate in the midstream of the
evolutionary flow. (Embedded) Simulation of the evolution-
ary flow, whose (upward) bending direction agrees with the
direction of the experimental observation.
been simulated by the linguistic model, which should be
intrinsic properties related to underlying grammars [18].
We also find the relationship between the average pro-
tein length and the ratio of amino acid frequencies. The
species of three domains gather in different regions in
the space of the average protein length and the ratio
RHQW/GV of average frequency for several later amino
acids (H, Q, W) to average frequency for several earlier
ones (G, V) [Fig. 4]. The points of all species form a
bending line [Fig. 4], which can be explained as an evo-
lutionary flow in that (i) the species with large (small)
genome locate in the midstream (margin) of the flow [Fig.
4] and (ii) the (rightward) evolutionary direction paral-
lels the directions of decreasing correlations of protein
length distributions among groups of the closely related
species. The evolutionary flow can be simulated by our
model [Fig. 4, Embedded]. The evolutionary direction
and the bending direction in the simulation agree with
the evolutionary flow of the 106 species.
In conclusion, the holographic bound improves our un-
derstanding of life, which supervises the maximum com-
plexity of life. Linguistics is necessary in storage of in-
formation in the protein/DNA sequences. We show that
the particular variance of amino acid frequencies and GC
content for the contemporary species are the products of
certain genetic code multiplicity and amino acid chronol-
ogy evolved in primordial time. The linguistic model
succeeds not only in the simulations of respective aspects
(amino acid frequencies [Fig. 1], GC content [Fig. 2b],
protein length distribution [Fig. 3]) but also in their re-
lationships (amino acid frequencies and GC content [Fig.
2a], amino acid frequencies and protein length distribu-
tion [Fig. 4]). So the thorough and detailed fit between
simulations and experimental observations confirms the
validity of the linguistic framework, which is grounded in
general principles in physics.
We thank Hefeng Wang, Liu Zhao, Donald R. Fors-
dyke, Zhenwei Yao for valuable discussions. Supported
by NSF of China Grant No. of 10374075.
∗ dirson@mail.xjtu.edu.cn
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mailto:dirson@mail.xjtu.edu.cn
http://cubic.bioc.columbia.edu/pep
