Viral RNA as a branched polymer

Myriad viruses use positive-strand RNA molecules as their genomes. Far from being only a repository of genetic material, viral RNA performs numerous other functions mediated by its physical structure and chemical properties. In this chapter, we focus on its structure and discuss how long RNA molecules can be treated as branched polymers through planar graphs. We describe the major results that can be obtained by this approach, in particular the observation that viral RNA genomes have a characteristic compactness that sets them aside from similar random RNAs. We also discuss how different parameters used in the current RNA folding software influence the resulting structures and how they can be related to experimentally observable quantities. Finally, we show how the connection to branched polymers can be extended to take advantage of known results from polymer physics and can be further moulded to include additional interactions, such as excluded volume or electrostatics.Comment: 24 pages, 9 figure

Branching analysis of viral RNA genomes

Strukturo dolgih molekul RNA, med katere sodijo tudi genomi virusov RNA, je eksperimentalno še zmeraj težko določiti, zato so za njihovo preučevanje neizogibne računske napovedi struktur. Napovedni algoritmi so sicer točnejši za krajše RNA, vendar omogočajo določanje razlik v strukturi razvejenosti in prostorske velikosti tudi med raznovrstnimi daljšimi RNA. Prav razvejenost strukture in prostorska kompaktnost pa pomembno vplivata na učinkovitost samosestavljanja virusov RNA. Osrednje vodilo dela je zato vprašanje, kako je globalna struktura RNA vsebovana v primarnem zaporedju. V delu predstavimo delovanje algoritmov za termodinamično napoved sekundarne strukture RNA in za statističnofizikalni opis uporabimo model razvejenega polimera. Analiziramo več kot 1700 genomov virusov RNA in na napovedanih strukturah izračunamo tipične topološke količine iz teorije grafov, s katerimi lahko ovrednotimo razvejenost in prostorsko kompaktnost strukture. Rezultate primerjamo z napovedmi za nabore naključnih zaporedij RNA, s čimer dobimo vpogled v morebitni evolucijski selekcijski pritisk, ki ohranja kompaktnost genomov ikozaedričnih virusov. Z dvema različnima pristopoma za naključno RNA ocenimo tudi skalirna eksponenta $rho$ in $varepsilon$, ki opisujeta skaliranje povprečne velikosti vej in povprečne dolžine poti v limiti velikih polimerov, ter pokažemo, da velja $rho simeq varepsilon approx 0.67$. Med vsemi znanimi modeli polimerov sta skalirna eksponenta naključne RNA najbližje modelu trirazsežnega samoizogibnega drevesa. Robustnost analize in izluščenih skalirnih eksponentov pokažemo z izbiro različnih naborov energijskih parametrov in naključnimi zaporedji z neenakomerno nukleotidno sestavo.The structure of long RNA molecules, such as genomes of RNA viruses, cannot be reliably determined experimentally, and hence any study of it inevitably involves computational structure predictions. Although the prediction algorithms are more accurate for short RNAs, they can be used to compare the branching structure and spatial size between various long RNAs. Branching pattern and compactness are also the key determinants of RNA virus self-assembly. The main aim of the Thesis is to clarify how the global structure of RNA is encoded in its primary sequence. We explain the core ideas behind algorithms for thermodynamic prediction of secondary structure of RNA, and use branched polymers as a model to describe the RNA. We analyse more than 1700 viral RNA genomes, and use the predicted structures to calculate the graph-theoretical topological measures of compactness. Results are compared to the predictions for sets of random RNAs, which elucidates the presence of evolutionary pressure which keeps genomes of icosahedral viruses spatially compact. Furthermore, we use two different approaches on random RNAs to calculate their exponents $rho$ and $varepsilon$, which describe the scaling of average branch weights and average path lengths in large-size limit, and we show that $rho simeq varepsilon approx 0.67$. Compared to other models of polymers, the scaling exponents for random RNA are most compatible with those of three-dimensional self-avoiding trees. Finally, we assess the robustness of estimated scaling exponents by using different sets of energy parameters and random RNAs with non-uniform nucleotide compositions

Modelling of cell growth

Celična rast je ena od ključnih lastnosti vsakega organizma. V diplomskem delu sem se osredotočil na rast modelnega prokariontskega organizma E. coli, tako da sem strnil ključna spoznanja o biokemijskih procesih, odgovornih za rast celice, in opisal tri različne grobe matematične modele za opis celične rasti. Prav tako sem se podrobneje posvetil modelu avtorjev Maitra in Dill, za katerega sem tudi ponovil numerične rezultate in izpostavil neskladnost z eksperimentalnimi podatki.Cell growth is one of the key signatures of living beings. In this thesis I have focused on the cell growth of prokaryotic organism E. coli. I have concisely described the most important biochemical processes that govern cell growth and described three different mathematical models that explain cell growth. Furthermore, I have repeated the numerical calculations of the model created by Maitra and Dill, and exposed some of the inconsitencies with the experimental data

Scaling properties of RNA as a randomly branching polymer

Formation of base pairs between the nucleotides of an RNA sequence gives rise to a complex and often highly branched RNA structure. While numerous studies have demonstrated the functional importance of the high degree of RNA branching -- for instance, for its spatial compactness or interaction with other biological macromolecules -- RNA branching topology remains largely unexplored. Here, we use the theory of randomly branching polymers to explore the scaling properties of RNAs by mapping their secondary structures onto planar tree graphs. Focusing on random RNA sequences of varying lengths, we determine the two scaling exponents related to their topology of branching. Our results indicate that ensembles of RNA secondary structures are characterized by annealed random branching and scale similarly to self-avoiding trees in three dimensions. We further show that the obtained scaling exponents are robust upon changes in nucleotide composition, tree topology, and folding energy parameters. Finally, in order to apply the theory of branching polymers to biological RNAs, whose length cannot be arbitrarily varied, we demonstrate how both scaling exponents can be obtained from the distributions of the related topological quantities of individual RNA molecules with fixed length. In this way, we establish a framework to study the branching properties of RNA and compare them to other known classes of branched polymers. By understanding the scaling properties of RNA related to its branching structure we aim to improve our understanding of the underlying principles and open up the possibility to design RNA sequences with desired topological properties.Comment: 13 pages, 4 figures; 10 pages and 8 figures of supplementary materia