Information and entropy

One of the most useful concepts in the physical sciences is that of entropy. To parameterize the thermodynamic descriptions of free energy and entropy (which is related to heat) to statistical physics, Gibbs developed the formula S $ =-{k}_B\sum_a{P}_a\log {P}_a $ , which describes the entropy in terms of the statistical probabilities $ {P}_a $ . In this expression, the Boltzmann constant $ {k}_B $ has units of energy per degree Kelvin. However, for general informatics, there are no inherent units associated with the entropy (i.e., $ {k}_B\to $ 1).

For a population that has completely adapted to environmental homeostasis, the distribution of genomic variants within that population optimizes the overall population health. That population will maintain this characteristic allelic distribution to promote its continuing survivability. Genomic variants of particular interest are single-nucleotide polymorphisms (SNPs). The overwhelming majority of SNPs are biallelic (i.e., express either of two allelic variants). The maintained order of the SNP variants can be quantified in terms of its entropy. For biallelic SNPs, the specific entropy of a given SNP labeled (S) is defined as:

(1) $$ {s}^{(S)}=-\sum_{a=1}^2{P}_a^{(S)}{\mathit{\log}}_2{P}_a^{(S)}, $$

where $ {s}^{(S)} $ is the specific entropy of the SNP, and $ {P}_a^{(S)} $ is the probability (which is related or frequency) of the occurrence of allele $ a $ in that population. The entropy is maximum for a completely stochastic SNP distribution (i.e., a population for which both alleles occur with equal probabilities), and it is zero for a homogenous SNP (i.e., only one allele is found among the whole population).

Biological correlations among a vicinity of SNPs manifest within fixed regions of the genome defined as haploblocks. Haploblocks are sets of SNPs with relatively few numbers of haplotypes associated with individuals. These haplotypes are transmitted between generations and likewise maintain their frequencies within a population in homeostasis. However, different populations have differing haploblock structures. For correlated SNPs in haploblocks, the haploblock specific entropy $ {s}^{(H)} $ is defined as:

(2) $$ {s}^{(H)}=-\sum_h^{2^{n^{(H)}}}{P}_h^{(H)}{\mathit{\log}}_2{P}_h^{(H)}, $$

where $ {n}^{(H)} $ is the number of SNPs in haploblock (H) and $ {P}_h^{(H)} $ is the frequency of occurrence of haplotype h in the population. The factor $ {2}^{n^{(H)}} $ represents the maximum possible number of haplotypes composed of biallelic SNPs. This identification relates to an SNP entropy as that of a haploblock containing one single SNP. As defined, entropy is an additive variable of the state of the statistical distribution. Therefore, the overall degree of disorder of the whole genome is quantified by the total sum of the specific entropies of both linked and non-linked SNPs to obtain the following:

(3) $$ {s}_{genome}=\left[\sum_H{s}^{(H)}+\sum_S{s}^{(S)}\right], $$

Whereas the entropy quantifies the degree of disorder within a statistical distribution, the information content $ \mathrm{IC}={s}_{max}-{s}_{genome} $ quantifies the degree of maintained order within the distribution. The entropy $ {s}_{max} $ represents the maximum possible whole genome entropy, which for biallelic SNPs is the total number of SNPs in the genome $ {n}_{SNPs} $ . Similarly, the minimum entropy is that of a completely homogeneous population, resulting in a vanishing entropy. Thus, the minimum IC is zero and the maximum is $ {s}_{max} $ . It is convenient to define the normalized information content (NIC) for the whole genome as:

(4) $$ {NIC}_{genome}=\frac{s_{max}-{s}_{genome}}{s_{max}}. $$

The NIC as defined varies from zero for any maximally disordered distribution to unity for any completely homogeneous distribution. Since it represents a unitless universal measure, the NIC can be used to compare genomes of widely disparate populations (even species), prose written in different languages, and even apples to oranges. For this reason, the NIC can be utilized to interrogate information from different regions of the genome, as well as from differing populations. Furthermore, it gives considerable insight into genomic variations without requiring foreknowledge of biological function or details of genomic history (Lindesay et al., Reference Liu, Yang, Ge, Liu and Zhao2012).

Information dynamics

Although an in-depth exploration of the informatics of a statistical distribution can be quite revealing of a multitude of important characteristics, it lacks the potential to explain the dynamics of the system being characterized (i.e., how the system changes and responds to stimuli). Biological groups, in particular, present very complicated systems with regard to modeling their information dynamics. For instance, optimizing population health in response to some environmental pathogens (like malaria and its vectors) involves the introduction of particular alleles that can affect the health of an individual in either beneficial or detrimental ways. However, sometimes for biallelic variants, one allele is highly advantageous within one extreme environment while the other is highly advantageous in the other extreme.

To quantify the information dynamics of a system, a universal dimensional unit has been introduced as a relative measure of how much of an effect a stimulus has on the system. The dynamic genomic unit should be associated with the degree of genomic variation of the shared variants. For genodynamics, the standard GEU is assigned to environmental agitations that invoke maximum variation of biallelic SNPs (i.e., alleles with equal frequencies) among the population for SNPs that are not in linkage disequilibrium.

Once a universal unit of variation has been ascertained, it can be used to compare the relative degree of influence of quantifiable external stimuli on the dynamic system. Genodynamics requires that genomic stimuli be quantified using a smoothly varying well-defined value $ {\lambda}_{stimulus} $ for each stimulus. Once the genomic potential $ {\mu}_a $ of allele $ a $ (which has units of GEUs) is defined, the adaptive force is then calculated as the gradient down the slope of that genomic stimulus:

(5) $$ {f}_a=-\frac{\partial {\mu}_a}{\partial \lambda }. $$

Genomic potentials will be assigned using guidance provided by energies and potentials defined in physical sciences. The genomic potentials should reflect that populations residing in environments with more vigorous pathogens and stimulants manifest higher degrees of variation when compared to those populations subject to fewer stimulants. When quantifying adaptive forces, it is important that the populations represented by the dynamic variables be in homeostasis with those environments. Although living systems are very far from thermodynamic equilibrium, a population in environmental homeostasis likewise displays a robust stability under environmental perturbations and fluctuations. In thermodynamics, systems in thermal equilibrium subject to a uniform external agitation (quantified by the temperature T) have their Helmholtz free energy F minimized. This means that this free energy remains unchanged under small fluctuations in quantified stimuli. (i.e., dF=0).

Variations in the Helmholtz free energy are typically expressed in terms of temperature T, entropy S, pressure P, volume V, and chemical potentials $ {\mu}_j $ associated with population numbers $ {N}_j $ , as given by the following:

(6) $$ dF=- SdT- PdV+\sum {\unicode{x03BC}}_jd{N}_{j.} $$

In thermodynamics, the temperature quantifies a universal set of environmental stimuli that parameterizes the overall degree of agitation of the system. In analogy, the genomic free energy of a population in homeostasis with its environment is parameterized by an environmental potential $ {T}_E $ that characterizes the overall degree of agitation and sustenance by the environment. It will be assumed that any ‘pressure’ that the population exerts upon the environment is negligible. For a given population, the genomic free energy is expressed as follows:

(7) $$ d{F}_{genome}=-{S}_{genome}\hskip0.5em d{T}_E+\sum_a\hskip0.4em {\mu}_a{dN}_a+\sum_h\hskip0.4em {\mu}_h{dN}_h, $$

where $ {\mu}^{(S)} $ is the genomic potential of SNP (S) for those SNPs that are not in linkage disequilibrium, and $ {\mu}^{(H)} $ is the genomic potential of haploblock (H). The genomic entropy of the population is defined as the size of the population times its specific entropy: S_genome=N_population×s_genome. The dynamic units (GEUs) are carried by the environmental potential and the genomic potentials, and the entropy and population numbers N are dimensionless.

The genomic energy of a homeostatic population should be an additive variable of state that depends on the genodynamic variables in Eq. (7). Each SNP potential is the population average of the two allelic potentials in that SNP, $ {\mu}^{(S)}\equiv \sum_a{\mu}_a^{(S)}{P}_a^{(S)} $ . Likewise, the haploblock potential $ {\mu}^{(H)} $ is the population average of its haplotype potentials. The population stability condition requires that the size of the population is stable (i.e., $ \frac{d{F}_{genome}}{d{N}_{population}}=0 $ ). Recognizing that the number of alleles $ a $ in the population is the frequency of that allele times the size of the population $ {N}_a^{(S)}={P}_a^{(S)}{N}_{population} $ , Eq. (7) becomes:

(8) $$ {\displaystyle \begin{array}{c}d{F}_{genome}=\left[-{s}_{genome}d{T}_E+\sum_S\sum_a\hskip0.4em {\mu}_a^{(S)}d{P}_a^{(S)}+\sum_H\sum_h{\mu}_h^{(H)}d{P}_h^{(H)}\right]\\ {}{N}_{population}+\left[\sum_S\sum_a\hskip0.35em {\mu}_a^{(S)}{P}_a^{(S)}+\sum_H\sum_h{\mu}_h^{(H)}{P}_h^{(H)}\right]d{N}_{population}.\end{array}} $$

The population stability condition then implies that the population averaged SNP potentials and block potentials sum to zero over the whole genome $ \left\langle {\mu}^{\left(\mathrm{all}\;\mathrm{S}\right)}\right\rangle +\left\langle {\mu}^{\left(\mathrm{all}\;\mathrm{H}\right)}\right\rangle =0 $ . This condition inherently incorporates Hardy–Weinberg equilibrium (Lindesay et al., Reference Murphy and Weaver2013; Alsufyani, Reference Alsufyani2019). The population thus maintains its allelic distribution throughout generations.

For a dynamic population, the allelic potential of a particular SNP should scale with the environmental potential. Differences between allelic potentials should reflect the frequencies of occurrence in an additive manner. These characteristics are incorporated in the dimensionless form:

(9) $$ \frac{\mu_{a2}^{(S)}-{\mu}_{a1}^{(S)}}{T_E}=-{log}_2\frac{P_{a2}^{(S)}}{P_{a1}^{(S)}}. $$

This expression clearly vanishes when each probability is $ \frac{1}{2} $ . As mentioned before, in this case of maximum variation, the allelic potential of each allele is assigned the value $ \overset{\sim }{\unicode{x03BC}}\equiv $ 1 GEU. This requires that each potential take the following form:

(10) $$ {\unicode{x03BC}}_a^{(S)}=\left(\tilde{\unicode{x03BC}}-{T}_E\right)-{T}_E\;{log}_2{P}_a^{(S)}. $$

A lower allelic potential is thus assigned to the allele that is more frequent (i.e., more conserved within the population). If the allele is homogenous throughout the population (i.e., $ {P}_a^{(S)}=1\Big) $ then that conserved allele is said to have a fixing potential given by $ {\unicode{x03BC}}_{fixing}\equiv \overset{\sim }{\unicode{x03BC}}-{T}_E $ . Similarly, the haplotype potential has the following form:

(11) $$ {\unicode{x03BC}}_h^{(H)}=\left(\tilde{\unicode{x03BC}}-{T}_E\right){n}^{(H)}-{T}_E\;{log}_2\;{P}_h^{(H)}. $$

The development of potentials with units allows modeling of the dynamics beyond just statistical informatics about the population as a whole. Furthermore, one can perform meaningful comparisons of the effect of an influence on differing populations with shared SNPs.

Performing the population averages on the genomic potentials from Eqs. (10) and (11), the population stability condition then determines the environmental potential $ {T}_E $ :

(12) $$ {\unicode{x03BC}}_{genome}=\left\langle {\unicode{x03BC}}^{\left( all\hskip0.24em S\right)}\right\rangle +\left\langle {\unicode{x03BC}}^{\left( all\hskip0.24em H\right)}\right\rangle =0\to {T}_E=\frac{\tilde{\unicode{x03BC}}\;{n}_{SNPs}}{n_{SNPs}-{S}_{genome}} = \frac{\tilde{\unicode{x03BC}}}{\;{NIC}_{genome}}, $$

where $ {\mathrm{n}}_{\mathrm{SNPs}} $ is the total number of SNPs on the genome. This form is analogous to the thermodynamic temperature of statistical physics in that more homogenous populations have lower environmental potential.

The data associating genomic variance with biological functions (e.g., genome wide association studies) often link phenotypes with individual SNPs. However, alleles that are in linkage disequilibrium mathematically manifest a maintained collective distribution defining a particular haplotype in a haploblock. Such collections of SNPs therefore have decreased entropy, which is reflected through lowered genomic potentials. It is therefore advantageous to distribute a haploblock’s potential $ {\unicode{x03BC}}^{(H)} $ among its constituent SNPs. The haploblock potential has been distributed using the following criteria (Lindesay et al., Reference Lindesay, Mason, Hercules and Dunston2018a):

• • if an allele is homogenous throughout the population, its distributed SNP potential is the same as the fixing potential $ {\unicode{x03BC}}_S^{(H)}=\tilde{\unicode{x03BC}}-{T}_E $ (equivalent to a non-linked SNP);

• • the distributed SNP potentials are additive and sum to the haploblock potential $ \left\langle {\unicode{x03BC}}^{(H)}\right\rangle =\sum_S\;{\unicode{x03BC}}_S^{(H)} $ ;

• • the haploblock potential is distributed among the SNPs proportionate with the degree of allelic variation within each given SNP.

The form of the distributed SNP potential that satisfies these criteria is given by the following:

(13) $$ {\unicode{x03BC}}_S^{(H)}={\unicode{x03BC}}_{fixed}+\left[{\unicode{x03BC}}^{(H)}-{n}^{(H)}{\unicode{x03BC}}_{fixed}\right]\left[\frac{{\overline{P}}_{s\prime }}{\sum_{s\prime }{\overline{P}}_{s\prime }}\right], $$

where $ {\overline{\mathrm{P}}}_{\mathrm{s}\prime } $ is the minor allele frequency of SNP $ \mathrm{S}^{\prime } $ . The haploblock has an increased degree of maintained order when compared to that of the individual SNPs. Therefore, each SNP manifests a binding potential given by the following:

(14) $$ {\varepsilon}_{binding}^{(s)}={\unicode{x03BC}}_S^{(H)}-\left\langle {\unicode{x03BC}}^{(s)}\right\rangle, $$

which is negative, thereby lowering the SNP potential. The definition of a binding potential $ {\varepsilon}_{binding}^{(s)} $ then also allows the distribution of the block potential to the individual alleles $ \mathrm{a} $ within the SNP,

(15) $$ {\unicode{x03BC}}_a^{(H)}={\unicode{x03BC}}_a^{(s)}+{\varepsilon}_{binding}^{\left(\mathrm{s}\right)}, $$

which is likewise lowered due to the collective behavior of the alleles.

• • Population-based values of environmental potentials:

Finally, the calculation of the various genomic potentials of a given population requires a determination of the overall degree of the environmental agitation of each population characterized by T_E. Prior investigations have indicated that the information content of the whole genome is very closely reflected by that of chromosome 3 (Alsufyani, Reference Alsufyani2019), which has therefore been used to calculate T_E for the various populations. Furthermore, the predictions of genodynamics most accurately describe adaptive forces for those population that have remained in environmental homeostasis for many generations. For this reason, data from populations presently residing within their ancestral geographical regions were selected. The populations examined include Peruvian in Lima, Peru (PEL), Colombian in Medellín, Colombia (CLM), Finnish in Finland (FIN), Kinh in Ho Chi Minh City, Vietnam (KHV), Japanese in Tokyo, Japan (JPT), Toscani in Italy (TSI), Mende in Sierra Leone (MSL), Han Chinese in Beijing (CHB), Iberian populations in Spain (IBS), and Yoruba in Ibadan, Nigeria (YRI). The environmental parameters here examined include virus, bacteria, helminth, and protozoa as zoonotic pathogens as well as chiroptera, primates, rodentia, and soricomorpha as zoonotic hosts. The calculated environmental potentials T_E for each population are indicated in Table 1 (Alsufyani, Reference Alsufyani2019).

Table 1.

Environmental potentials of the examined populations

Viral zoonotic pathogens

The variant rs1010211 (with common alleles T and C) flagged for an adaptive dependency due to viral pathogen among bats, carnivores, hoofed mammals, moles, primates, rodents, and shrew (Han et al., Reference Hong, Piao, Sun, Tao and Ke2016). The functional behaviors of the SNP potential and that of its alleles are demonstrated in Figure 1.

Figure 1.

(a) The correlation between the single-nucleotide polymorphism (SNP) rs1010211 and richness pattern of viral pathogens. (b) The correlation between allele C of rs1010211 and richness pattern of viral pathogens. (c) The correlation between allele T of rs1010211 and richness pattern of viral pathogens (Alsufyani and Lindesay, Reference Alsufyani and Lindesay2022).

In the figures, the vertical axes represent the respective genomic potentials in units of GEUs, and the horizontal axes are expressed in units of viral richness. As defined in the data source (Han et al., Reference Hong, Piao, Sun, Tao and Ke2016),

‘Richness: the number of unique species within a particular geographic area; richness is a count-based metric for quantifying diversity, which contrasts with other metrics, such as functional trait diversity (the different types of traits represented within a geographic area) or genetic diversity.’

The SNP potential plotted in Figure 1a indicates a positive adaptive force of about +0.06 GEU’s/viral zoonosis unit with a relative RMS uncertainty of 0.065. Figure 1b indicates a direct positive adaptive force on the C allele of about 0.04 GEU’s/viral zoonosis unit with relative RMS of 0.056. This allele becomes nearly homogeneous for the populations residing within regions of highest viral zoonoses. In contrast, the T allele plotted in Figure 1c encounters a negative adaptive force of more than −0.2 GEU’s/viral zoonosis unit with relative RMS uncertainty of 0.031. This functional form indicates a direct genomic reaction via the frequency of occurrence of this allele within the population. It should be noted that this SNP is not in linkage disequilibrium for any of the populations considered.

The intron variant rs1010211 lies within the TNIK gene, which is a component of the adaptive immune response. This ancestral variant is common in zoonotic mammals (TRAF2 and NCK interacting kinase, 2023). TNIK is evolutionarily conserved among amphibians, aves, mammalia, and ray-finned fishes. More than 197 organisms share orthologs retaining the function of human TNIK. This gene also regulates cell division and cell death (Shkoda et al., 2012). TNIK activates B-cells, which function in the humoral immunity component of the adaptive immune system (Murphy and Weaver, Reference Rusan, Li and Hammerman2016), producing specialized antibody molecules that then serve as B-cell receptors (Alberts et al., Reference Alberts, Johnson, Lewis, Raff, Roberts and Walter2002). Recently, TNIK has been found to be a regulator of effector and memory T cell differentiation by inducing a population of undifferentiated memory T cells (Jaeger-Ruckstuhl et al., Reference Lindesay, Mason, Hercules and Dunston2020). Therefore, the fact that this variant responds to viral zoonotic adaptive forces is thus confirmed by biophysical quantification.

The TNIK gene map is illustrated in Figure 2.

Figure 2.

The map of the TRAF2 and NCK Interacting Kinase (TNIK) locus. Chromosome 3 has 199 million base pairs. The gene containing the flagged single-nucleotide polymorphism (SNP), TNIK, extends from 171,058,414 to 171,460,408. The SNP rs1010211 is at locus 171,413,851 (Alsufyani and Lindesay, Reference Alsufyani and Lindesay2022).

TNIK is a member of the germinal center kinase (GCK) family (Yu et al., Reference Han, Kramer and Drake2014). Germinal centers are transient structures within B lymphocytes that adapt their antibody genes during the immune response to an infection (Natkunam, Reference Savci-Heijink, Halfwerk, Koster, Horlings and Van De Vijver2007). These centers play a crucial role in the adaptive humoral immunity component that generates matured B cells, producing effective antibodies against infectious agents. They also play a role in the production of durable memory B cells (Yin et al., Reference Yu, Zhan, Wang, Zhang and Chen2012). It should be noted that GCKs are also involved in innate immune regulation.

Rodents zoonotic pathogens

The intron variant rs16864017 (with common alleles T and C) flagged an adaptive dependency due to rodent zoonoses. The adaptive behaviors of the alleles are demonstrated in Figure 3.

Figure 3.

(a) The dependence of allele T of rs16864017 upon richness pattern of rodent pathogens. (b) The dependence of allele C of rs16864017 upon richness pattern of rodent pathogens.

In the figures, the vertical axes represent the respective genomic potentials in units of GEUs, and the horizontal axes are expressed in units of zoonotic richness in rodents.

The T allele in Figure 3a experiences a positive adaptive force of about 0.12 GEU’s/rodent zoonosis unit, with relative RMS deviation of 0.046. The collective linked behavior of this allele displays a highly favorable genomic potential for the populations most exposed to these pathogens. On the other hand, the C allele plotted in Figure 3b is most conserved for the populations that are least exposed to these pathogens, displaying a negative adaptive force of about −0.17 GEU’s/rodent zoonosis unit, with relative RMS uncertainty of 0.081. This indicates that the increased presence of rodent zoonosis negatively affects the favorable distribution of the C allele among the population. It should be noted that this SNP is in linkage disequilibrium for all of the populations considered. This suggests a collective biological function shared between this set of SNPs in linkage disequilibrium.

The intron variant rs16864017 lies within the gene TPRG1. The TPRG1 gene map is illustrated in Figure 4.

Figure 4.

The map of the tumor protein P63 regulated 1 (TPRG1) locus. Chromosome 3 has 199 million base pairs. The gene containing the flagged single-nucleotide polymorphism (SNP), TPRG1, extends from 188997227 to 189325304. The SNP rs16864017 is at locus 189146927.

This gene is differentially expressed in tumor tissues and has been reported to be involved in the regulation of the immune response (Liu et al., Reference Natkunam2019). TPRG1 has orthologs in several chordates, which includes all mammals (and specifically rodents) (Tumor Protein p63 Regulated 1, Reference Uffelmann, Huang, Ns, Vries, Okada, Martin and Posthuma2023). Knockdown (i.e., temporarily disabling or weakening the expression) of this gene has been found to suppress inflammation in rats with cystitis (bladder inflammation), as well as reduce cell proliferation and migration of human primary glandularis cells (Hong et al., Reference Kelley, Flam, Izumchenko, Danilova, Wulf and Guo2022). It should be noted that several human infections are known to be due to exposure to the urinary and/or respiratory aerosols of rodents (e.g., Hantavirus, Staphylococcus aureus, Streptococcus pneumoniae) (Williams et al., Reference Yin, Shi, Jiao, Chen, Wang, Greene and Zhou2008). TPRG1 has been found to stimulate inflammatory responses (Liu et al., Reference Natkunam2019). The expressions of this immune-related gene have also been correlated with early tumor recurrence (Wang et al., Reference Wang, Zhou, Wu, Liang, He and Peng2021). The allele-specific expression (ASE) of TPRG1 has been identified as having tumor-specific expression within the cancer genome atlas as well as in biological evaluation (Th et al., Reference Hardy2020). Furthermore, the ASE in this gene has been found to be specific to human papillomavirus (HPV) positive tumors (Rusan et al., Reference Sella2015; Kelley et al., Reference Lindesay, Mason, Hercules and Dunston2017; Corces et al., Reference Corces, Granja, Shams, Louie, Seoane and Zhou2018). TPRG1 was also found to be differentially expressed in HPV-associated oropharyngeal squamous cell carcinoma (Th et al., Reference Hardy2020) and breast cancer (Savci-Heijink et al., Reference Shkoda, Town, Griese, Romio, Sarioglu, Th and Kieser2019). It is possible that tumorigenesis is associated with chromosomal rearrangement of TPRG1 in normal lipoma tissues (Wang et al., Reference Williams, Elizabeth and Barker2010). Finally, it has been suggested that the antisense strand of this gene TPRG1-AS1 is involved in the suppression of liver cancer growth (Choi et al., Reference Choi, Kwon, Moon, Yoon and Shah2021).