A deterministic oscillatory model of microtubule growth and shrinkage for differential actions of short chain fatty acids

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Colorectal cancer (CRC) is the fourth most common cancer and second leading cause of cancer-related death in the UK.¹ Short chain fatty acids (SCFA) are closely associated with colon epithelia as by-products of anabolic fermentation by gut bacteria.² The principle SCFAs in the human gut are acetate, propionate, butyrate and valerate. These differ in carbon chain lengths (2, 3, 4 & 5 carbons, respectively). Butyrate is the most researched to date for colon health. It is the preferred energy source for the colonocyte, a potent regulator of cell fate,³ particularly apoptosis,^4,5 and is thought to be chemopreventive. SCFAs have been shown to induce acetylation of target proteins, either via histone deacetylation^6,7 or by shifting the balance of acetyl-CoA reactions towards acetylation;^6,7 however, the mechanistic details of their functions in colonocytes are unclear. Our previous empirical work and others^6,8 suggested that carbon chain length may impact on their actions, with odd and even carbon-length SCFAs possessing distinct functions by virtue of different pathways in â-oxidation and entry into the TCA cycle.

Microtubules (MT) are cytoskeletal proteins that perform many critical cellular functions. In addition to maintaining cell structure, motility, intra- and intercellular transport, they are central to accurate formation and control of the mitotic spindle during cell division. MT fibres are composed of typically 13 protofilaments arranged longitudinally, each of which is constructed from αβ-tubulin heterodimers arranged in a head-to-tail manner. This confers the MT fibre with polarity so that polymerisation primarily occurs at the plus end. During assembly, the α-tubulin subunits are bound within the fibre and inaccessible to external proteins; conversely, the β-tubulin subunits are exposed on the surface enabling them to undergo post-translational modifications (PTM) and interact with microtubule associated proteins (MAP), including dynein and kinesin motor proteins. The arrangement of β-tubulin isotypes along the fibre combined with their PTMs has been hypothesised to generate a tubulin code that may direct MT functions and regulate MT behaviour.^9,10 In order to perform their molecular functions MTs require the ability to rapidly grow, shrink and change direction in response to cellular cues. These transitions are an essential characteristic of MTs and are termed dynamic instability. The accepted view of MT dynamic instability is the MT capping model.¹³ Briefly, phosphorylated αβ-tubulin dimers bind to the plus end of the MT fibre to form a protective tubulin-GTP cap. Immediately following polymerisation, the β-tubulin-GTP subunits undergo irreversible hydrolysis to their GDP form. Although this  provides the necessary energy to drive MT growth,¹⁴ β-tubulin-GDP is unstable and the fibre will rapidly dissociate if the protective GTP cap is not sustained. This is termed catastrophe and occurs when the pool of αβ-tubulin-GTP subunits in the vicinity of the fibre tip falls below a critical concentration.¹⁵ Once released into the cytosol, β-tubulin-GDP subunits are free to rephosphorylate, restoring the critical concentration and enabling MT growth to resume, termed rescue. These actions are illustrated in Figure 1A. Experimental evidence on MT dynamics has primarily been derived from fluorescent tracking microscopy of growing and shrinking MTs in vitro.^12,16-19 The life history of an MT within a cell can range from several minutes to hours and involves extended periods of slow growth, short periods of rapid shrinkage and time spent pausing. Each period of elongation and shortening can consist of multiple catastrophes and rescues (Fig. 1B).^12,19 Despite appearing random, MT dynamic instability is a tightly regulated and orchestrated process.

Polymerisation of a β-tubulin GTP subunit to an MT fibre and its subsequent hydrolysis is a cooperative procedure. Suggested mechanisms include chemical and protein-protein interactions between β-tubulin GTP and GDP subunits;²⁰ conformational changes arising from GTP hydrolysis;²¹ and facilitated diffusion. The latter mechanism postulated that αβ-tubulin dimers are transported towards the plus end of an MT fibre by virtue of fibre’s polarity; as such, the probability that a dimer encounters an MT will increase as the MT lengthens.²²

The β-tubulin superfamily comprises of six principle classes thought to have functional significance.^23,24 Most β-tubulin isotypes are species, tissue or cell type specific. Their relative proportions within an MT fibre can be regulated by upstream signalling events, suggesting they may influence MT functional activity.¹¹ Consequently, disrupting the fine balance of β-tubulin isotypes can lead to aberrant tubulin behaviour and may have implications in the progression of colonic diseases, including CRC.^11,12

Our proteomic analyses investigated proteins differentially regulated in colon cancer cells by treatment with butyrate, propionate and valerate. The results identified several cytoskeletal proteins that were significantly dysregulated:²⁵ all three SCFAs targeted keratin 19; butyrate and propionate targeted actin; and propionate primarily targeted keratins 8 and 18; however, only propionate and valerate significantly targeted β-tubulin isotypes β2c-, β3- and β1-tubulin. These findings were consistent with our high content analyses (HCA) that had revealed valerate and propionate were the most effective in inducing MT breakdown and G2/M cell cycle arrest, despite being weaker effectors in apoptotic functions.^5,26 Whereas butyrate induces cytoskeletal breakdown via apoptosis,^4,5 these observations suggested that propionate and valerate may act through different mechanisms. Based on this evidence, it was hypothesised that odd- and even-chain SCFAs possessed distinct and unique metabolic functions.

SCFAs act by promoting post-translational acetylation of target proteins. Bioinformatic searches of Reactome and SABiosciences^11,27,28 indicated that the activities of several transcriptional regulators (TR) associated with β2c-, β3- and β1-tubulin genes (RFX, PPAR, NSFR and NF-κB) could be altered by acetylation. Histone acetylation also induces chromatin remodelling to promote transcription of target genes and can itself be influenced by upstream acetylation events.^29,30 These results suggested that treatment of colonocytes with SCFAs altered the regulation of specific genes, including overexpression of β2c- and β3-tubulin and suppression of β1-tubulin transcription via acetylation of NF-κB.^31,32 Both β2c- and β3-tubulin have been implicated in MT destabilisation;¹¹ although less is known about the role of β1-tubulin in MT dynamics.

Computational modelling has contributed to a better understanding of MT functions^21,33-36 and the biophysical details of MT dynamic instability. Various approaches have been adopted, including Monte Carlo simulations;^21,33 simplified coarse-grained stochastic models;³⁵ one- and three-dimensional modeling;³⁷ and models focusing on different types of lattices and MT fibre-fibre interactions.^33,36 The term “structural plasticity” invokes mechanisms more dependent on the structural conformations of bound subunits than their chemical states.²⁰ Simulated systems of dynamic MTs have also advanced understanding of their associations with MAPs, the collective behaviour of MTs and motors, and have been used to replicate mitotic events.^38,39 The model of Bayley et al. (1990) was pivotal in demonstrating the validity of the lateral-cap model, in which â-tubulin-GTP assembly is a cooperative stochastic event that is fully coupled with â-tubulin hydrolysis.⁴⁰ Their hypothesis was supported by Pedigo and Williams (2002) who showed that conformational changes arising from â-tubulin-GTP hydrolysis directly after association was best described using kinetic principles of lattice interactions.²¹ Computational models have been proposed as a basis for predicting the actions of antimicrotubule drugs (AMDs).^40,41 This had particular relevance for our investigation which aimed to explore the links between SCFAs, transcriptional regulation of â-tubulin isotypes, MT integrity and performance, and the potential use of SCFAs in adjuvant therapies in CRC.

A common approach to modelling systems of this nature is to use ordinary differential equations (ODEs). These present biological systems as continuous and deterministic and assume that randomness is not an important factor in the system of interest. An alternative approach is to use stochastic models if there is deemed to be inherent variability within the biological system; a stochastic approach will be used in scenarios whereby a small number of molecules are suggested to be involved in discrete random collisions, such as during the expression of a single gene. The main limitation of stochastic models is that they have a tendency to be computationally intensive, as reviewed in McAuley et al. (2013).^42-44 In this study, we sought to develop a minimal deterministic model based on experimental evidence that might explain or validate our hypothesis:²⁵ that different SCFAs imposed differential effects on the balance of β-tubulin isotypes and their impact on MT integrity. A comparison between the stochastic behaviour of MTs in vivo and in vitro and a deterministic representation in silico is presented in Figure 1B and C.