Abstract
Cell mechanosensing is implicated in the control of a broad range of cell behaviours, with cy-toskeletal contractility a key component. Experimentally, it is observed that the contractility of the cell responds to increasing substrate stiffness, showing increased contractile force and changing the distribution of cytoskeletal elements. Here we show using a theoretical model of active cell contractility that upregulation of contractility need not be energetically expensive, especially when combined with changes in adhesion and contractile distribution. Indeed, we show that a feedback mechanism based on maintenance of strain energy would require an upregulation in contractile pressure on all but the softest substrates. We consider both the commonly reported substrate strain energy and active work done. We demonstrate substrate strain energy would preferentially select for the experimentally observed clustering of cell adhesions on stiffer substrates which effectively soften the substrate and enable an upregulation of total contractile pressure; while localisation of contractility has the greatest impact on the internal work. It is well established that cells sense, adapt and respond to the mechanical properties of their environment. This mechanosensing is key across cell behaviours ultimately affecting e.g. cell growth, development and differentiation [1–3]. Fundamental to mechanosensing are contractile forces generated by myosin motors within an actin rich network in the cell. These forces are transmitted from the cell to the extracellular matrix through adhesions [4]. A focus of mechanotransduction research has traditionally been these sites of cell adhesion, which in stiffer environments, are concentrated into small patches of strong attachment called focal adhesions [1, 3]. Experimental investigations commonly use engineered gels or micropillar arrays with defined mechanical properties to observe cell response [5]. As well as changes in signal transduction, changes in structural elements associated with cellular contractility are observed, including increased actin density and stress fibre formation [6, 7]. Considering contractile forces, myosin and motor activity has been found to be more broadly distributed on soft gels becoming more localised, eventually overlapping with a dense actin cortex on stiffer substrates [8]. Hence it is typically observed that contractile forces increase with increased gel stiffness [9]. The mechanism by which stiffness changes lead to changes in contractile force is unclear although target stress or strain states are often implicated [10, 11]. To quantify the mechanical activity of cells on two-dimensional substrates different approaches have been suggested. Most commonly the applied tractions are measured, through e.g. traction force microscopy, and used to infer activity. Recent work has suggested that the substrate strain energy could be effectively used as a measure for overall mechanical activity [12, 13], leading to the observation that cells may respond directly to changes in substrate strain energy [14]. Substrate strain * c.dunlop@surrey.ac.uk energy has also been observed to be approximately conserved across a range of stiffnesses [15], suggesting mechanical feedback to actively control strain energy. Even without fully constraining the strain energy, it is clear that there are bounds on the energy budget [16], with a link between cell contractility and energy consumption demonstrated [17]. Theoretically models have explored this energy budget using energy constraints as drivers of differentiation [18] or cell shape control [19]. We here use an active matter model to investigate both the substrate strain energy and the work done by the active cell components. We show that over a broad range of stiffnesses upregulation of active contractile pressure does not require an increase in energy expenditure. Indeed, upregulation of contractility is compatible with constant strain energy. We also investigate the localisation of con-tractility into the cell cortex showing that this will have minimal effect on the substrate strain energy at realistic levels of cell adhesion, although the active strain energy is sensitive to these changes. This is consistent with the observation that localisation of contractility can lead to large internal strains [20]. Introducing localised patches of adhesions we see that these generate polarised cells, with clustering of adhesion points particularly energetically favourable in terms of substrate strain energy, thus enabling significantly higher total contractile pressure. Active Matter Model. Active matter models consider the cell as an elastic material with an additional component of stress generated by active contraction. Active contraction may be modelled either through computational simulations of cytoskeletal filaments [21–23] or via a continuum approach [24–26]. We adopt the continuum approach taking the stress within the cell σ = σ P + σ A , where σ P is the passive cell elasticity and σ A is the active stress generated by the contractile pressure. Noting that the timescale for cell adhesion is faster than the relaxation timescale viscoelastic effects may be neglected and we assume a linear elastic response in both the substrate and the cell, see e.g. [26]. Furthermore, as the dimension of the spread cell is greater than its height h we consider