This paper uses a novel approach to study change in the rapidly evolving and globally accessible TrEMBL protein databases available at 1. The TrEMBL protein databases accumulate the sequenced proteins which result from the efforts of countless teams of researchers around the planet. After more than 25 years of growth, they are already extremely large and still growing rapidly with the most recent release at the time of writing being release 2024 02 dated 27-Mar2024 of UniProtKB/TrEMBL which contains 248,234,451 sequence entries, comprising 87,367,689,973 amino acids. The proteins vary in length from the shortest A0A0G2JLF7 HUMAN at just 7 amino acids all the way up to a staggering 45,354 amino acids in the longest currently known, A0A5A9P0L4 9TELE. Clearly these enormous numbers are effectively intractable in terms of identifying local patterns, or even phylogenetically shared patterns but the discipline of statistical physics over the last 150 years from its origins in the kinetic theory of gases in the hands of the visionary physicist Ludwig Boltzmann, has produced techniques for dealing with such extraordinarily large numbers. Compared with the number of molecules in 1 cubic meter of gas at standard temperature and pressure (2.68 × 10²⁵), even the TrEMBL databases pale into insignificance. In spite of these vast numbers, the methodology of statistical physics comfortably handles them automatically aggregating any and all local mechanisms to go directly to the equilibrium or most likely distribution. In the case of a gas, the velocities all asymptote to the Maxwell-Boltzmann distribution 2.

By incorporating information theory into this methodology, 34 it was shown that all discrete systems (systems composed of countable pieces) sharing only the property that their individual pieces are distinguishable and requiring no other commonality, exhibit patterns dominated by a power-law. These patterns arise from a conservation principle, CoHSI or the Conservation of Hartley-Shannon Information. If we consider the most basic discrete system that consists of ordered strings (components) of coloured beads, then from CoHSI the theoretically predicted length distribution for any discrete system looks like Figure 1. The presence of this distribution has been identified now in a wide variety of dissimilar discrete systems - lengths of words in texts, number of words in sentences in texts, in large collections of software irrespective of their language or functionality and most notably for the purposes of this paper, in proteins 3. The prior knowledge that such an equilibrium distribution of lengths is present in any substantial collection of proteins guides our novel approach allowing us to measure significant departures from that equilibrium state and understand the reasons for any such departures and this we now do.

Proteins are an exemplary discrete system, in that we can consider them as strings of amino acids, the length of a protein being measured by the total number of amino acids. The TrEMBL database1 represents essentially the totality of our knowledge of the structure and diversity of proteins. One might expect that the distribution of protein lengths would be shaped by natural selection acting on particular structure/function properties, but whereas individual proteins will be subject to natural selection in the normal way, the overall distribution of the properties of proteins results from the complex interactions of many processes; natural selection, genetic drift, random extinctions etc. CoHSI, because of its statistical mechanical framework 3 aggregates all these mechanisms and predicts a scale-free equilibrium outcome that is simply the overwhelmingly most likely state. The theoretically predicted length distribution for any discrete system looks like Figure 1.

Thus it was predicted (and borne out experimentally) that the length distributions of proteins would show the scale-free distributions implied by CoHSI 345. Prior to the Covid-19 pandemic we can clearly see the predicted CoHSI distribution in protein lengths, for example in the 2017 TrEMBL release 17-03 (we consider other releases of TrEMBL as this narrative develops) as Figure 2a - Figure 2b. The similarity between the CoHSI prediction in Figure 1 and the data of Figure 2a is compelling both visually and statistically.

Figure 2b is a cumulative complementary distribution function 6, a widely used noise-suppressing display of the same data as Figure 2a. The left hand (y-)axis is the number of proteins longer than the size

shown on the x-axis. On the left hand side, it is flat corresponding to the sharp rise to the peak of Figure 1. Reading off this plateau height on the y-axis gives the total number of proteins considered here, (just under 1 × 10⁸in release 17-03). As we move right and the length increases, fewer and fewer proteins are greater than this length and the data becomes the classic straight line on a log-log scale indicating the presence of the predicted power-law. The mere presence of a straight line is only a necessary condition for a power-law. For greater statistical confidence, a sufficiency test must also be run 78. Details of this are given in 3 where an emphatic power-law is confirmed .

In essence the above development establishes Figure 2b as an equilibrium distribution, an emergent property shared by all discrete systems 4.In the parlance of statistical mechanics, the mathematical framework behind CoHSI, the biochemical properties of the individual amino acids in the global system of proteins are irrelevant; proteins can be considered as simply consisting of strings of distinguishable amino acids 3. This pattern of proteinlengths shown in 2b is by definition an equilibrium distribution and for such a large distribution, we would normally expect little to disturb thisequilibrium. However, there was extraordinary activity in protein discovery, focused on SARS-CoV-2 that took place in 2019 and the years following as a result of theCovid19 pandemic; in the following section we explore the impact of this on the equilibrium distribution of protein lengths.

It is reasonable to ask why power-laws, as described here in the impacts of the Covid-19 pandemic are essentially ubiquitous in the natural world. As reviewed in detail in 4, power-laws are observed in phenomena as diverse as wealth and the frequency of word use, from the size of computer programs to the quantity of alcohol consumed, and from the growth of oyster shells to the size of craters on the moon. Logically, there are only two possibilities. Either there is a single underlying principle that generates power-law behaviour in complex discrete systems, or there are many specific local mechanisms that coincidentally generate the same outcomes, i.e. power-law distributions, in very different systems. CoHSI theory 34 provides a resolution of these two explanations; complex discrete systems are mechanistically complicated (and often seemingly random) but regardless of any and all mechanisms, distributions dominated by power-laws are the essentially inevitable equilibrium state of these systems.

No human subjects, human tissues or animals were used in this research.

This study adheres to the transparency and reproducibility principles espoused by 141516171819 and includes references to all methods and source code necessary to reproduce the results presented. For this study, the methods and source code are included in the wider set of reproducibility deliverables, available at https:// datadryad.org/stash share/9nVGYwauP_wdFM84hA6GlC52t4pFircx4NAGl_ukbYA. Each reproducibility deliverable allows all results, tables and diagrams to be reproduced individually for that study, as well as performing verification checks on machine environment, availability of essential open-source packages, quality of arithmetic and regression testing of the outputs 20.

No supplementary materials directly accompany this paper.

LH performed the analyses, LH and GW developed the arguments, discussed the results and contributed to the text of the manuscript. Both authors gave final approval for publication.

No institutional or external funding for this research was received by the authors.

The authors acknowledge the many researchers with whom they have discussed the implications of CoHSI in biological systems over the years, most notably the late Professor Bob Chapman who gave freely of his insights and vast experience.