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Introduction
Influenza A viruses have been responsible for four pandemics of severe human respiratory disease over the last century,
resulting in tens of thousands of deaths all over the
world[1-5]. Since 1997, avian influenza virus infections in poultry have
taken on new significance, with increasing numbers of cases involving bird-to-human transmission and the resulting
production of clinically severe and fatal human
infections[6-8]. At present, two out of three general conditions for the onset of a
pandemic have been met; namely, the emergence of a new virus and its ability to replicate in humans causing serious illness.
Should the virus achieve efficient human-to-human transmission, the next influenza pandemic might
occur[9].
By accumulating point mutations (genetic drift), the influenza A viruses change their antigenic properties mainly in the
RNA genes, which code for ten
proteins[10]. Hemagglutinin is one of the antigens for neutralizing antibodies and is involved
in the binding of virus particles to receptors on host
cells[11,12]. It plays an important role in the propagation
of the virus and in the infection of the
host[13,14]. There are two mechanisms to explain the host-mediated variation of influenza A viruses. The
first mechanism is the pressure of the
antibody[15]. The second is the selective pressure for the appearance of the host cell
variant with altered receptor binding
specificities[16].
The sequence analysis is widely used to estimate the mutations of influenza
viruses[17-20]. Other approaches have been
used to study the mutations of influenza viruses, such as the model of protein
evolution[21], the mathematical model capturing
both realistic epidemiological dynamics and viral evolution at the sequence
level[22], the travelling waves in a
one-dimensional model[23], and so on.
From 1999 to 2002, we have developed two computational mutation approaches to analyze the protein primary structure,
which include (i) the calculation of predictable and unpredictable portions of amino-acid pairs in a
protein[24-38], and (ii) the calculation of amino-acid distribution rank in a
protein[39-43]. We reviewed these approaches and their implications in
2002[44] (Table 1).
Since then, we have mainly been applying these approaches to analyze mutations in different proteins. One of the
advantages of our approaches is that we can use a single value as a numerical measure to represent a protein, and then
compare different proteins among a protein
family[45-49], evaluate the mutation effect on
proteins[50-63], reveal the mutation trend in
proteins[50-65], calculate the mutation
periodicity[66,67], and trace the mutation process along the time
course[66,67]. In this review we summarise our findings and their implications for mutation features of influenza A virus hemagglutinins in
combination with some newly available data.
Historical trend of hemagglutinin mutations
With our first approach (Appendix), we mainly use the unpredictable portion of amino-acid pairs to study the historical
trend of hemagglutinin mutations because the unpredictable portion is not engineered by randomness and is deliberately
constructed. Also it can present precisely a hemagglutinin with a single value, which is an unambiguous record of
hemagglutinin evolutionary process as we can view each hemagglutinin as a sample from its evolution.
Figure 1 displays the general mutation trend of influenza A virus hemagglutinins from 1918 to 2005 using the
unpredictable type and frequency of amino-acid pairs. Each symbol represents the mean value of unpredictable portion of all
full-length hemagglutinins in the given year. Several points can be drawn in Figure 1: (i) Two regressed lines indicate that the
unpredictable portions of amino-acid pairs are decreasing over time; that is, the hemagglutinins are structurally becoming
more stable and less sensitive to mutation. (ii) Any spike in fluctuated unpredictable portions suggests that the
hemagglutinin has experienced mutations at some time point, which may result in the influenza pandemic/epidemic. (iii) The statistical
means and standard deviations for these data are 63.32%±3.28% for unpredictable types and
70.46%±
2.78% for unpredictable frequencies, thus their standard
deviations correspond to around 13 type mutations (3.28%/
0.25%) and 15 frequency mutations
(2.78%/0.18%)[66].
With our second approach (Appendix) and similar rationale, we use the amino-acid distribution rank to study the
historical trend in hemagglutinin mutations. Figure 2 shows the general mutation trend of influenza A virus hemagglutinins from
1918 to 2005 using the amino-acid distribution rank. The regressed lines indicate the trend of the amino-acid distribution rank
with respect to different species over time. This measure is quite stable for the human hemagglutinins, which suggests that
the functional clusters in human hemagglutinins are relatively stable over time because any permanent re-distribution of
amino acids would lead to the change in its regressed line. On the other hand, the amino-acid distribution rank increases in
avian and equine hemagglutinins and decreases in swine hemagglutinins over time, which indicates that these
hemagglutinins have experienced the re-distribution of amino acids. As a result, the function of these hemagglutinins has either
increased or decreased over this period of time; that is, the hemagglutinin structure of influenza A viruses becomes more
probabilisti-cally complicated for avian and equine, but less complicated for
swine[67].
Periodicity of hemagglutinin mutations in the past
The fluctuated data in Figures 1 and 2 imply that there may be some kind of periodicity in mutation sensitivity of influenza
A virus hemagglutinins. Figure 3 displays the periodicity obtained from 1963 to 2005 using the fast Fourier transform
(Appendix), where each stick represents a periodicity and the stick height is the magnitude of change with respect to
unpredictable type/frequency of amino-acid pairs and amino-acid distribution rank. For example, the first prominent peak in
the unpredictable type of amino-acid pair is located at 2.3 years/cycle with the height of 2.24%, which means the
hemagglutinins would experience 9 type mutations (2.24%/0.25%=8.96, Appendix) at an interval of 2.3 years. This observation is
different from the findings in our previous
study[66], where we found 5 type mutations based on the data from 1971 to 2004.
This difference, of course, results from different databases as more data are documented herein, however it further supports
that the number of hemagglutinin mutations is indeed decreasing over time.
Notably, the prominent peak in the unpredictable frequency of amino-acid pair is located at a cycle of 11 years, which
could approximately be connected to the interval of three pandemics (the second pandemic in 1957, the third in 1968 and the
fourth in 1977[68, 69]).
Compared to the periodicities based upon unpredictable type/frequency of amino-acid pairs and amino-acid distribution
rank[67], we understand that there would be type mutations in quite a short period of time with the change in distribution of
functional clusters, then there would be frequency mutations in a relative long cycle. In other words, some types of
amino-acid pairs in functional clusters appear or disappear during a short period of time, thereafter the
mutations are more likely to modify the existing types of amino-acid pairs.
Mutation trend in different hemagglutinin subtypes
As can be seen in Figure 1, the historical trend of hemagglutinin mutations is that the unpredictable portions are decreasing.
Thus we would deduce that the larger the unpredictable portion is, the larger the mutation trend is, and this deduction is
identical to the findings in our studies on other
proteins[50-63].
Figure 4 illustrates the unpredictable portions of amino-acid pairs and distribution rank of amino acids in different
hemagglutinin subtypes from influenza A viruses. Although each hemagglutinin subtype has different sensitivity to
future mutations, a slight trend can be found in the top panel, for example, the unpredictable type decreases from H2 to H4,
from H5 to H7, and increases from H11 to H14. We are particularly interested in H1, H2, H3, H5, H7, and H9 because they are
linked to previous human infections. It is important to note that the unpredictable type of H5 hemagglutinins is remarkably
larger than others, so the H5 hemagglutinins have a stronger mutation trend than
others[46,66], we would therefore expect to
see more mutations in H5 hemagglutinins until their unpredictable type reaches the average
level, which is currently about 63.3%. This means that we would anticipate
12 type mutations in the future (3%/0.25%) seeing that the mean value of
unpredictable type is about 66.5% in H5 hemagglutinins.
However, H5 hemagglutinins do not appear remarkably larger than the others in unpredictable frequency and distribution
rank (the middle and bottom panels in Figure 4). As mentioned in the Appendix, the unpredictable type includes both absent
and present types of amino-acid pairs, whereas the unpredictable frequency and distribution rank deal only with the amino
acids present in a protein. Thus, the unpredictable type of amino-acid pair does indicate the mutation trend for the future, as
some absent types can appear through mutations.
Mutation potency in different species
Although the percentage of unpredictable portions indicates the mutation trend, the unpredictable portions include a
large number of amino-acid pairs, for example, the hemagglutinin of 1918 "Spanish" influenza virus contains 565 amino-acid
pairs, of which 434 are unpredictable (76.81%). To search the amino-acid pair with big mutation potency, we use the
difference between actual and predicted frequencies (Appendix) to determine the amino-acid pairs sensitive to mutations
because this difference can serve as a measure of the structural stability of an amino-acid pair; that is, the smaller the
difference the more stable the construction of the amino-acid pair. In particular, the larger the positive difference the more
unpredictable the present amino-acid pair and, in contrast, the larger the negative difference the more unpredictable the
absent amino-acid pair. In practice, the amino-acid pairs whose actual frequency is larger than their predicted one are likely
to be targeted by mutations. In contrast, the amino-acid pairs whose actual frequency is smaller than their predicted one are
more likely to be formed through mutation according to our previous
studies[50-62].
Hence, the hemagglutinin containing a large number of amino-acid pairs with a big difference between actual and
predicted frequencies is more sensitive to mutations than that with a large number of amino-acid pairs with a small difference
between actual and predicted frequencies. Figure 5 gives us an idea of such an analysis, where two aspects can be seen. (i)
The bars are not symmetric with respect to the zero difference (x-axis), indicating that there are more amino-acid pairs with a
positive difference and less with a negative difference. In general, there are more amino-acid pairs being targeted by mutation
than those formed through mutation. (ii) The avian hemagglutinins have more amino-acid pairs with larger positive
differences than human ones. For instance, the largest positive difference is up to 9 in avian, but only 6 in humans. Seeing
that the species vulnerability depends on the number of amino-acid pairs with larger differences, there would be more
mutations in avian, this elucidates why so many mutations have been found in avian
influenza viruses, and supports the notion that the avian species harbour a large reservoir of influenza virus
strains[70-75].
In addition, we can compare the mutation trend among different birds by reason that they are directly correlated to the
occurrence of bird flu. Figure 6 shows that there are more unpredictable types in aquatic avian than in terraneous ones, thus
water flock, especially aquatic birds and geese, are more sensitive to
mutations[66]. These features provide the evidence
supporting the hypothesis that aquatic birds are the primordial source of all influenza viruses in other
species[1,76].
Outlook of hemagglutinin mutations for the future
Because unpredictable portions of amino-acid pairs and their fluctuation are becoming smaller along the time course
(Figure 1), the trend line and channel clearly suggest the future of influenza virus hemagglutinins (Figure 7). Generally
speaking, we would expect that the unpredictable portions and their fluctuation are continuing smaller along the trend line
and channel although some fluctuations may go beyond the band. In such a case, there would be fewer pandemics in this
century than in the last one. This presumption can be supported by the fact that there were four major influenza
epidemics/pandemics recorded during the less than 20 years between 1830 and
1848[77]. It was quite the opposite, with only four
influenza pandemics recorded in the last century.
These phenomena can be explained by our
approach[44-62,66]. The unpredictable amino-acid pairs should deliberately be
conserved for a certain purpose because their construction requires more time and energy, so that a protein maintains only
absolutely necessary numbers of unpredictable amino-acid pairs. During the evolutionary process, nature is trying to
minimize the unpredictable portion through mutations, which may bring about new unpredictable amino-acid pairs, and the
newly introduced mutations result in the
fluctuations[66].
Our position on the current cycle of hemagglutinin evolutionary process
Based on the fast Fourier transform of unpredictable type of amino-acid pairs, we can approximately designate the
hemagglutinin evolutionary process with a cycle of around 7 years, and furthermore we can estimate our position at the
current cycle of hemagglutinin evolutionary process to determine how many years remain before the next spike, which may
be related to severe mutations. Figure 8 demonstrates our position at the current cycle of hemagglutinin evolutionary
process, and we are approaching the last year of the cycle. Compared with five historical lines, the unpredictable type of
amino-acid pairs has a 3/5 chance of going up.
Figures 9 and 10 display our position at the current cycle of hemagglutinin evolutionary process based on the fast Fourier
transform of unpredictable frequency of amino-acid pairs and amino-acid distribution rank, from which we can approximately
elucidate the hemagglutinin evolutionary process with a cycle of around 11 years. We have three years before finishing this
cycle, however the historical lines suggest the possibility of a spike.
Probability of occurring of mutation in hemag-glutinins
The undetermined factor for future mutation is the impact, which can either lead to the mutation or has no effect,
meanwhile a mutation can occur without any impact (Appendix). Figure 11 shows the change in amino-acid distribution rank
of different hemagglutinin subtypes from 1992 to 2002. Although we have yet to know what the impact was in 1997, the impact
did interrupt the pattern of amino-acid distribution ranks in different subtypes of hemagglutinins along the time course.
Nevertheless, the hemagglutinin of influenza A virus must have the ability to mutate itself, otherwise no impact would
have an effect on it. This ability is the probability of the occurrence of mutation, which can be determined using the
cross-impact analysis with Bayesian equation (Appendix). Table 2 shows the probability of the occurrence of mutation in different
subtypes of hemagglutinins and neuraminidases. As can be seen in Table 2, the H5 hemagglutinins have the maximal of 0.427
chance of occurrence of mutation without any impact, which is the largest probability of occurrence of spontaneous
mutations among different hemagglutinin subtypes. Also, the chance of occurrence of spontaneous mutation is larger in N1
neuraminidases than in N2 ones[65]. Thus, the H5N1 viruses have the largest chance of simultaneous occurrence of
mutations among different subtypes of hemagglutinins and neuraminidases, which can explain why there are so many mutations
in H5N1 influenza viruses[72-76].
In fact, the mutation chance listed in the lower part of Table 2 is quite small if we consider that these probabilities represent
the largest chance of simultaneous occurrence of mutation. This means that the impact does play an important role in the
enhancement of the occurrence of mutation. Thus, we need to implement the probability of occurrence of mutation
P(1) into a dynamic frame; that is, the chance of occurrence of mutation with different intensities of an impact. This is defined by the
cross-impact analysis as the impacted probabilities
P(1/2¯) and P(1/2) with equations 1 and 2 (Appendix).
Figure 12 illustrates the impacted probabilities with respect to the probability of occurrence of mutation
P(1) and to the probability of occurrence of impact
P(2) in different hemagglutinin subtypes from influenza A viruses, which can be read as
follows. First, the left panels show these three probabilities in a 3-dementional configuration: the
P(1) in x-axis, the
P(2) in z-axis and the impacted probabilities in y-axis. Second, both impacted probabilities
P(1/2¯) and P(1/2) are represented by two
triangles ABC and ACD, respectively. Third, they are transferred into a 2-demensional figure (black and gray triangles in the
right panels), so the dynamics of impact on the occurrence of mutation can be viewed easily and
clearly[65].
Three clues to insights can be drawn from Figure 12: (i) as the intensity of the impact
P(2) increases, probability of the occurrence of spontaneous mutations
P(1|2¯) decreases(black triangle ABC), whereas the probability of the occurrence of
P(1/2) induced mutations P(1/2) increases (gray triangle
ACD); (ii) the impact can significantly enhance the probability of the
occurrence of mutations P(1), whose range enlarges to the top-right corner from bottom-left; for example, the
P(1) in H1 hemagglutinins changes to the range of
0.653-1 from the range of 0-0.347 (top panel); and (iii) among different
hemagglutinin subtypes, the probability of the occurrence of mutations (both the black and gray triangles) is large in H5
hemagglutinins, which further supports the view that H5 hemagglutinins mutate more frequently. Thus, Figure 12 provides
a quantitative way to predict the mutation trend of influenza virus hemagglutinins if we could predict an impact and define its
intensity.
Direction of our future studies
In this review, we summarise the results at the first stage of our
studies[46,48,65-67,78]; that is, we focus mainly on the general
trend of hemagglutinin mutations. This is not only because we are still developing our methods, but also because at this
stage the systematic trends are more important than the trends in an individual case, which is masked by numerous random
and uncertain factors. We hope to understand how the principal average or trend arises before we are able to explain what is
causing individual cases to depart from it.
In our future studies, one direction is to monitor the change in unpredictable portions and distribution rank with new
available data, since we should be alert to a possible outbreak of influenza and bird flu if a significant change is recorded. The
other direction is to predict future mutations, because given a mutation in a protein, our first approach reveals the mutation
trend of an individual amino-acid pair; our second approach indicates the sensitive position, and our third approach, which
is the translation probability between RNA and mutated amino
acid[79], highlights the possible mutated amino acid. Together
with these three approaches, we may predict the mutations at a specific amino acid in a particular region of influenza virus
hemagglutinins in the future.
Appendix
Methodology
The rationale and detailed description of our approaches have been published in all of our previous
publications[24-67,78]; however, as they
are not yet familiar to many researchers, we describe our methods with the example of the hemagglutinin of 1918 "Spanish" influenza A virus
(accession number AF117241), which is the earliest complete sequence documented in the data
bank[80,81].
Amino-acid pair predictability
As we know that an amino-acid pair in a protein is composed of any 20 kinds of amino acids, so theoretically there are 400 possible types
of amino-acid pairs. In terms of amino-acid pairs, the distinguishing of protein differs in the number of possible types of amino-acid pairs
and/or in the frequency of each type. The 1918 hemagglutinin is composed of 566 amino acids, thus there are 565 amino-acid pairs. Of 400
possible types, 137 are absent and 263 present: 111 types appear once, 71 twice, 42 three, 24 four, 8 five, 5 six, 1 nine and 1 eleven times,
respectively.
Randomly predictable present type of amino-acid pair with predictable frequency
There are 44 glycines (G) and 37 threonines (T) in the 1918 hemagglutinin, the frequency of random presence of amino-acid pair GT is 3
(44/566×37/565×565=2.876); that is, GT would
appear three times in the protein. In fact, we do find three GT in this hemagglutinin, so the actual frequency of GT is 3. In this case both the
presence of the type GT and its frequency are predictable, and the difference between its actual and predicted frequencies is 0.
Randomly predictable present type of amino-acid pair with unpredictable
frequency There are 51 leucines (L) in the 1918
hemagglutinin, the frequency of random presence of amino-acid pair LL is 5
(51/566×50/565×565=4.505); that is, there would be five LL in
this hemagglutinin. But actually the LL appears eleven times, so the presence of LL is predictable, but its frequency is unpredictable, and the
difference between its actual and predicted frequencies is 6.
It is also the case that the actual frequency is larger than the predicted one. Another case is that the actual frequency is smaller than
predicted. For instance, there are 35 glutamic acids (E) in the protein, the predicted frequency of EG is 3
(35/566×44/565×565=
2.721), while its actual frequency is only 1 and the difference between its actual and predicted frequencies is -2.
Randomly unpredictable present type of amino-acid pair
There are 13 histidines (H) in the 1918 hemagglutinin, and the predicted frequency of HH is 0
(13/566×12/565×565=0.276), so the type HH would not appear in this hemagglutinin. However it appears twice in the reality, thus the presence of HH is unpredictable. Naturally its
frequency is unpredictable too, and the difference between its actual and predicted frequencies is 2.
Randomly predictable absent type of amino-acid pair
There are 8 methionines (M) and 17 glutamines (Q) in the 1918
hemagglutinin. The predicted frequency of MQ is 0
(8/566×17/565×565=
0.240); that is, the MQ would not appear in this hemagglutinin, which is true in the real situation. Thus the absence of MQ with its frequency
is predictable, and the difference between its actual and predicted frequencies is 0.
Randomly unpredictable absent type of amino-acid pair
There are 49 serines (S) and 37 alanines (A) in the 1918 hemagglutinin.
The predicted frequency of SA is 3
(49/566×37/565×565=3.203); that is, there would be three SA in this hemagglutinin. However, in reality
no SA appears, therefore the absence of SA from this hemagglutinin is unpredictable. Naturally its frequency is unpredictable too, and the
difference between its actual and predicted frequencies is -3.
Predictable and unpredictable portions of amino-acid
pairs
After the calculations described above, the amino-acid pairs in a
protein can be classified as predictable and unpredictable portions with respect to type and frequency, and the sum of both predictable and
unpredictable portions is 100%. Of the amino-acid pairs in the 1918 hemagglutinin, the unpredictable type and frequency are
68.75% and 76.81%, respectively. Either the predictable or unpredict-able portions can serve as a quantitative measure to present a protein.
Difference between actual and predicted frequencies of
amino- acid pairs
The unpredictable amino-acid pairs can be further divided
into two parts: one is the actual frequency smaller than predicted, while the other is the actual frequency larger than predicted. To compare
the constructive stability of amino-acid pairs, we calculate the difference between actual frequency and predicted frequency of amino-acid pairs
in a protein. Considering 400 theoretical amino-acid types of the 1918 hemagglutinin, 153 types (38.25%) have the actual frequencies smaller
than predicted and the average difference is -1.183, while 122 types (30.50%) show the actual frequencies larger than predicted and the average
difference is 1.492. Taking 565 amino-acid pairs of the 1918 hemagglutinin into account, 71 pairs (12.57%) contain actual frequencies
smaller than predicted and the average difference is -1.197, whereas 363 pairs (64.25%) reveal the actual frequencies larger than predicted and
the average difference is 1.846.
Type mutation and frequency mutation
As there are 400 types of theoretically possible amino-acid pairs and we use the 100% to
classify them as predicable and unpredictable types, thus 0.25% represents one of 400 types, so a 0.25% change indicates that one of 400 types
mutates to an unpredictable type from a predictable type or vice versa. This is the type mutation. However, the situation related to the
frequency of amino-acid pairs is dependent on the length of amino-acid sequence. Hence, approximately 0.18%
(1/565) change can be regarded as a modification in an amino-acid pair in the 1918 hemagglutinin, as there are 565 amino-acid pairs in the
protein. This is the frequency mutation.
Amino-acid distribution probability/rank
The position of any 20 kinds of amino acids in a protein can be determined by experimental approach, so each kind of amino acid has a
certain distribution pattern in a protein with respect to its position. Furthermore a certain distribution pattern can be associated with a certain
probability, which can be calculated according to the occupancy problems of subpopulations and
partitions[82]. For a certain distribution of a
kind of amino acid in a protein, its distribution probability is equal to
r!/(q0!×q
1!×...×qn!)×
r!/(r1!×r2
!×...×rn!)×n
-r, where ! is the factorial function;
that is, n!=n×(n -
1)×(n - 2)×...×1, 0!=1 by definition,
r is the number of a type of amino acid,
q is the number of parts with the same number
of amino acids and n is the number of grouped parts in the protein for a type of amino acid.
For example, there are eight metheonines (M), the least abundant amino acid, in the 1918 hemagglutinin, how do these 8 M distribute
among 566 amino acids in this hemagglutinin? We can imagine to group this hemagglutinin into 8 parts, and each one contains about 71 amino
acids (566/8=70.75). Table 3 lists all 22 possible distribution patterns regarding 8 M in 8 parts and their distribution probabilities and ranks.
The first eight columns in Table 3 show that the 1918 hemagglutinin is grouped into 8 parts, and the first eight cells in each row represent a
possible distribution pattern of M, and the last two columns display the corresponding distribution probability and rank.
As different distribution patterns can have the same distribution probability, we rank the distribution probabilities according to a
descending order, thus the largest distribution probability is ranked as one. Again in our example, there are 22 possible distributions for 8 M
in 8 parts in Table 3, while there are only 18 distribution ranks. In general, the smaller the distribution rank is, the larger the distribution
probability is. Although there are many possible distributions for a type of amino acids in a protein (such as 22 possible distributions for 8 M),
the protein in question possesses only one distribution pattern, therefore there is only one distribution probability/rank for each kind of amino
acid and a maximum of 20 distribution probabilities/ranks in a protein.
Similarly, we imagine to group 11 parts for 11 tryptophans (W), 13 parts for 13 histidines (H), 16 parts for 16 cysteines (C), 17 parts for
17 glutamines (Q), 19 parts for 19 phenylalanines (F) and prolines (P), 20 parts for 20 arginines (R), 25 parts for 25 aspartic acids (D), 26 parts
for 26 tyrosines (Y), 32 parts for 32 isoleucines (I) and valines (V), 33 parts for 33 lysines (K), 35 parts for 35 glutamic acids (E), 37 parts
for 37 alanines and threonines (T), 42 parts for 42 asparagines (N), 44 parts for 44 glycines (G), 49 parts for 49 serines (S), 51 parts for 51
leucines (L) in the hemagglutinin of 1918 "Spanish" influenza virus, and conduct the similar calculations.
As different kinds of amino acids have different contributions to a protein, we standardize them by means of the distribution rank per
amino acid, which is calculated by dividing the rank of each kind of amino acids by the number of corresponding amino acids. In the 1918
hemagglutinin, the distribution rank of metheonines is 1/8=
0.125, because these 8 metheonines distribute in the hemagglutinin with the largest probability (0.2523). Accordingly, the sum of ranks for
all 20 kinds of amino acids is 22.8698, thus the distribution rank in this protein is 22.8698/20=1.1435. Naturally, the distribution rank can
serve as a quantitative measure to present a protein.
Implication of our approaches
The construction of a protein with large predictable portions of amino-acid pairs and with small amino-acid distribution rank is certainly
a way to adapt to the fast-changes in surroundings and environments, as the speed of construction of a protein might be crucial for its survival,
this would require the least time and energy.
However, nature should deliberately construct the amino-acid pairs whose actual frequency differs from the predicted frequency and whose
amino acids cluster somewhere of a protein with a larger amino-acid distribution rank. The functional amino-acid pairs should be deliberately
evolved, so a protein keeps only absolutely necessary unpredictable amino-acid pairs. During the evolutionary process, nature is trying to
minimize the unpredictable portion through mutations, which may bring about new unpredictable amino-acid pairs, so the evolutionary process
is continuing.
Fast Fourier transform
One of important applications of Fourier analysis is to determine the periodicity in a chaotic fluctuating dataset, so we use it to analyze
the potential periodicity of hemagglutinin mutations over
time[66,67].
Cross-impact analysis
An impact can either lead to the mutation or has no effect, meanwhile a mutation can occur without any impact. In the context of our
study, these relationships can be represented by means of the cross-impact analysis in Figure
13[83-88]. At the impact level,
P(2¯) is the probability of not occurring of an impact, and the
P(2) is the probability of occurring of an impact. At the mutation level, the
P(1|2¯) is the impacted probability of occurring of mutation without an impact, the
P(1¯|2¯) is the impacted probability of not occurring of
mutation without an impact, the
P(1¯|2) is the impacted probability of not occurring of mutation with an impact, and the
P(1|2) is the impacted probability of occurring of mutation with an impact. At the influenza level, both
P(1¯2¯) and P(1¯2) are the probabilities of not occurring of
influenza without and with an impact, whereas both
P(12¯) and P(12)are the probabilities of occurring of influenza without and with an impact.
The impact on mutation can be traced through Figure 13, for example, the probability
P(12) will increase if an impact enhances the
chance of the occurrence of mutation. Most importantly, Bayes¡¯ law indicates that the probabilities of occurrences of two events can be
related by
where P(1) is the probability of the occurrence of mutation. In the cross-impact analysis, both enhancement and inhibition have been defined
with respect to coupled events. The enhancement is that the occurrence of the second event enhances the probability of occurrence of the
first event; that is,
P(1|2)>P(1), in our case, an impact increases the occurrence of mutation. The inhibition is that the occurrence of second
event inhibits the probability of the occurrence of the first event; that is,
P(1|2)<P(1), in our case, an impact decreases the occurrence of
mutation. When we are dealing with the situation that the impact enhances the mutations, the following equations are
used[87,88]:
What Figure 13 says is that each event has two complementary probabilities, namely, occurrence and non-occurrence, and influence and
non-influence on the following event. In more complicated forms, the impact can be many, and each has its occurrence and non-occurrence, and
influence and non-influence. However, we cannot directly use the analysis in Figure 13 and equations 1 and 2 because we know neither what
is/are the impact/impacts leading to the mutations in proteins of influenza A viruses nor how large is the probability of spontaneous mutations
in them. In order to predict the mutation trend in different hemagglutinin subtypes, we conducted a study with the combination of Bayes¡¯
law[65].
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