## Abstract

The mechanical response of single cells and tissues exhibits a broad distribution of time scales that gives often rise to a distinctive power-law regime. Such complex behaviour cannot be easily captured by traditional rheological approaches, making material characterisation and predictive modelling very challenging. Here, we present a novel model combining conventional viscoelastic elements with fractional calculus that successfully captures the macroscopic relaxation response of epithelial monolayers. The parameters extracted from the fitting of the relaxation modulus allow prediction of the response of the same material to slow stretch and creep, indicating that the model captured intrinsic material properties. Two characteristic times can be derived from the model parameters, and together these explain different qualitative behaviours observed in creep after genetic and chemical treatments. We compared the response of tissues with the behaviour of single cells as well as intra and extra-cellular components, and linked the power-law behaviour of the epithelium to the dynamics of the cell cortex. Such a unified model for the mechanical response of biological materials provides a novel and robust mathematical approach for diagnostic methods based on mechanical traits as well as more accurate computational models of tissues mechanics.

As part of their physiological function, single cells and tissues are continuously exposed to mechanical stress. For example, leukocytes circulating in the blood must squeeze through small capillaries, and the epidermis must deform in response to movements of our limbs. During development, mechanical forces initiate morphogenetic processes involving epithelial growth, elongation or bending, acting as cues to coordinate morphogenetic events [1]. Epithelial cell sheets are also continuously subjected to deformation as part of normal physiology. For instance, lung epithelial cells are exposed to fast cyclical mechanical stress during respiration [2], while epithelia lining the intestinal wall or those in the skin can experience long lasting strain [3]. Failure to withstand physiological forces results in fracture of monolayers which may lead to severe clinical conditions, such as hemorrhage or pressure ulcers [3]–[6]. Despite significant progress with the experimental characterization of cell and tissue mechanics, understanding the role of mechanical forces in development and pathology is hampered by the lack of a unified quantitative approach to capture, compare and predict the complex mechanical behaviours of tissues, cells, and sub-cellular components across all physiologically relevant time-scales. Such a framework would also enable us to assess the effects of pharmacological treatments on tissue mechanical response without necessitating experimental characterization of the tissue response to all loading conditions, something important for tissue engineering and the design of palliative treatment strategies.

In recent years, experimental characterization of the mechanical behaviour of single cells and tissues has revealed a complex set of mechanical behaviours in response to deformation [7]–[10]. For example, both single cells and tissues often display multiphasic responses in stress relaxation and creep tests, which comprise a combination of power-law and exponential behaviours. Power-law responses are commonly observed in biomaterials and are thought to originate from their complex hierarchical structure [11]–[14]. These behaviours cannot be easily modelled using traditional linear viscoelasticity, where constitutive rheological models result from combinations of elastic springs and viscous dashpots that translate into sets of linear ordinary differential equations [9], [15]–[18]. In this framework, power laws can only be implemented through a large numbers of linear elements [19], making this approach impractical and uninformative. Empirical functions have been introduced to overcome this challenge [11], [20], [21], but the lack of underlying material model, in the form of a computable differential equation, prevents the direct comparison of data collected under different loading conditions.

One potential approach for modelling the mechanics of materials presenting power-law behaviours is fractional calculus [22]. This relies on the introduction of a mechanical viscoelastic element called a spring-pot whose behaviour is intermediate between a spring and a dashpot [23]. This element based on fractional derivatives captures, with only two parameters, the broad distribution of characteristic times [24] typical of the mechanical response of cellularised materials. This element has recently been combined with traditional elements to model more complex rheological behaviours, referred to as generalized viscoelastic models [25].

In this paper, we examine the potential of generalized viscoelastic models for modelling biological materials by combining traditional rheological elements with the springpot. With only four parameters, we capture the time-dependent response of single cells and epithelial monolayers. Using parameters extracted from relaxation tests, we are able to predict the response of the same material to creep and ramp deformations with no further fitting, and relate the model parameters to single cell characteristics as well as recent measurements of cortical rheology.

## A constitutive model for epithelial monolayers

One widely used model system for studying tissue mechanics is the epithelium monolayer. We focused on MDCK cell monolayers devoid of substrate, of typical width of 2 mm and suspended between two rods at a distance of 1.5 mm. Despite the absence of a substrate, cells still retain epithelial characteristics [9]. Such a material has now been extensively studied [26], [27], and the effect of pharmacological treatments on rheological properties characterized [20]. The advantage of such a simplified system lies in the fact that the tension is transmitted only through the intercellular junctions and the cytoskeleton, but not the extracellular matrix. The relaxation response (response to a step in strain, see experimental details in section 1 in Supplementary materials) consists of an initial power-law phase in the first 5 s, followed by an exponential phase that reaches a plateau at ∼60 s (figure 1) [20]. This description establishes the minimum number of parameters needed to describe such a rich behaviour: (i) the level of the final plateau, (ii) the exponential decay time, (iii) the power-law exponent and (iv) the transition time between the two phases. This qualitative analysis will now inform the development of a novel rheological model tailored to capture these four components of the response using the minimum number of parameters.

A branch of Mathematics called Fractional Calculus provides conceptual and numerical tools well suited to capture power law behaviours [28], [29]. In traditional calculus, a function can be differentiated *n* times, where *n* is an integer. For viscous (fluid-like) materials, the stress is proportional to the first time derivative of the strain, where *n* = 1. For elastic (solid-like) materials, the stress is proportional to the strain, which can be seen as the zero-th time derivative of the strain *n* = 0. Fractional calculus generalizes the differentiation process such that the number *n* can now be real (see Supplementary materials section 2). With the spring-pot fractional element, the stress is proportional to the *β* derivative of the strain, where 0 ≤ *β* ≤ 1:
where *c*_{β} is a constant dependent on the material and *d*^{β}/*dt*^{β} is the fractional derivative operator. When *β* = 0, the material behaves like a spring, and, when *β* = 1, like a dash-pot. As *β* varies from 0 to 1, the response of the material continuously transitions from elastic to viscous behaviour and if a step change in stress or strain is applied, the response exhibits a power-law. Mathematically, the response is only defined by an integral over time, leading to strong history dependence, referred to as the hereditary phenomena (Supplementary materials section 2). Despite this complexity, the spring-pot still lies in the linear viscoelastic framework, enabling us to greatly simplify the analysis of the data and make predictions. For dimensional consistency, the unit of the constant *c*_{β} is (Pa s^{−β}), and therefore it does not have a straightforward physical meaning, although, it has been argued that it may represent a measurement of the firmness of the material [30].

The spring-pot can be combined with other rheological elements to generate a rich set of behaviours [31]. Configurations explored so far were mostly selected for their mathematical simplicity, rather than relevance to particular physical systems [25], [32]. Here, we adopt a phenomenological approach based on our qualitative description of the material’s behaviour, aiming to capture both its short and long time-scale response. At long time scale, the stress response shows a plateau (figure 1 (a)). Hence the model requires a spring in parallel with a dissipative branch that would not carry any tension in steady state. At intermediate time scales, the stress relaxes exponentially towards the plateau (figure 1 (b)). Hence the dissipative branch should behave as a dash-pot in series with an element that would transiently store deformation, for instance a spring as in a Maxwell model [33]. However, at short time scale (0.1-10 s), the dissipative branch exhibits a power-law. The presence of a power-law relaxation response immediately after application of strain, rather than a specific jump in stress, indicates that we should replace the spring in the dissipative branch with a spring-pot (see figure 2 (a)). The fractional model introduced in the dissipative branch is a special case of a known combination referred to as a Fractional Maxwell Model (FMM) [31]. The constitutive equation for the fractional material model introduced here in figure 2 (a) is reported in the Supplementary materials section 2. In what follows, such a qualitative approach is validated against experimental data, leading to a predictive model of the material’s behaviour.

## The generalized fractional viscoelastic model characterizes the biphasic stress relaxation response

The relaxation modulus (stress response to a unit strain of deformation) of the fractional network model introduced above can be derived analytically. Since the relaxation modulus of two elements in parallel is given by the sum of their relaxation moduli, the relaxation modulus *G*(*t*) of the novel viscoelastic model presented in figure 2 (a) is obtained by adding the relaxation modulus of the Fractional Maxwell Model (FMM) to the stiffness of the spring, which results in
where *E*_{a,b} (*z*) is the Mittag-Leffler function, a special function that arises from the solution of fractional differential equations (see Supplementary materials section 2). The qualitative behaviour of the relaxation modulus is plotted in log-log scale in figure 2 (a). Since the argument of the Mittag-Leffler function is non-dimensional, we can identify a first characteristic time *τ*_{1}, given by which approximates the transition time between the power-law and the exponential regime. The significance of *τ*_{1} as a characteristic relaxation time will be evident when the model is used to analyze real data.
To assess the validity of the fractional model, we used it to fit the relaxation response of epithelial monolayers (see Supplementary materials section 1 for details of the experimental set up). In agreement with the qualitative analysis of the curves (figure 1), the four parameters involved in equation 2 account for the experimental data (see figure 2 (b)), successfully capturing all time domains: the power-law regime (for t < 10 s), the exponential behaviours and the steady-state stress. We further examined if the model could capture the relaxation response of epithelial sheets in which myosin contractility, one of the most important component controlling cellular mechanical properties, was inhibited [34]. We observed indeed that the same four parameters could fit well the experimental data (figure 2 (c)). This allows us to compare the parameters extracted from treated monolayers with their control (DMSO treated, figure S6 in Supplementary materials). The viscosity *η* of treated monolayers doubles compared to the DMSO treated monolayers (figure S6 (c) in Supplementary materials), in line with cell-scale findings suggesting that dynamic contraction of actin filaments increases cell fluidity [7]. By contrast, a reduction of the stiffness *k* was observed (figure S6 (d) in Supplementary materials), which suggests that acto-myosin contractility mainly plays a role in stress dissipation at long time-scale, consistent with the conclusions previously presented by [7], [20]. Other treatments have been applied to monolayers to examine the role of actin network organization and crosslinkers, without observing significant variations in the relaxation response of monolayers (see figures S6 and S7 in Supplementary materials).

## The generalised fractional viscoelastic model predicts the response to different loading conditions with no further fitting

The model relies on the assumption that the material behaves linearly. To identify the linear domain, we examined the stress response at different strain amplitudes ranging from 20% to 50%. We can observe that the material parameters are almost constant until roughly 30% (see figure S8 in Supplementary materials), which provides an upper bound of the linear domain where we expect the model to be valid. Within this range, we can assess the predictive power of our rheological description of epithelial monolayers. We extracted a distribution of parameters from the stress relaxation data and used them to estimate the response of the material to different forms of mechanical stimulation. Good agreement between predictions and experiments over a broad range of testing protocols would signify that our description represents a constitutive model whose parameters can be seen as material properties. We first consider the stress response to a strain ramp applied at constant strain rate (1%/s). The predicted response for the untreated monolayers is shown in figure 3 (a), with 95% confidence interval (see sections 3 and 5 Supplementary materials for details about prediction and statistical analysis, and see figure S5 (a) for Y27632 treated monolayers). The experimental results and the analytical predictions are in excellent agreement with no free parameters.

Similarly, we can challenge the model by predicting and validating the deformation response *J* (*t*) of the epithelial monolayers to a unit step in stress, a test usually referred to as a creep experiment (see section 3 Supplementary materials). For linear viscoelastic materials the relation between relaxation and creep moduli in the Laplace domain is relatively simple, and given by After transforming the relaxation modulus in equation (2) in the Laplace domain, we find:
To obtain the solution in the time domain *J* (*t*), the inverse Laplace transform of the equation above is performed numerically.

The creep response is richer than the relaxation response, for which *k* only added a simple offset to the stress. Here, because the imposed load can continuously redistribute between the two branches of the model, *k* is involved in the dynamics. We can indeed identify in the creep response an additional time-scale *τ*_{2}involved in the response: We therefore have one additional dimensionless parameter which controls the shape of the creep response *ξ τ*_{1},/*τ*_{2}, The value of *ξ* leads to qualitatively different responses as plotted in figure 3 (b). (i) If *ξ* < 1, at short times we first observe a power-law behaviour arising from the spring-pot followed by an exponential regime where the dashpot dominates. The transition from spring-pot-dominated to dashpot-dominated regime is governed by the characteristic time *τ*_{1}, as for the relaxation response. While the deformation increases, the spring eventually becomes relevant and the system tends towards the plateau as a Kelvin-Voigt model with a characteristic time *τ*_{2}. (ii) If *ξ* > 1, the spring saturates before the transition to the dashpot occurs in the dissipative branch. Hence, the model behaves as a Fractional Kelvin-Voigt model with a characteristic time which can be expressed as is irrelevant in this regime. (iii) If *ξ* ≈ 1 the transition from the spring-pot to the dashpot corresponds to the time at which the spring becomes relevant.Therefore, the transition from the spring-pot to the Kelvin-Voigt model occurs with characteristic time *τ*_{1} ≈ *τ*_{2}.

The model is now used to predict the response of monolayers when subjected to a step in stress using the material parameters derived from the relaxation experiments. We performed new experiments to test our model’s predictive power (see section 1 in Supplementary materials). In these experiments, we subjected monolayers to a stress which was maintained constant after an initial short ramp in strain (see inset figure 3 (c)). Strikingly, the experimental data falls well within the 95% confidence interval of the predicted response with no free parameters (details in section 3 and 5 in Supplementary materials) (figure 3 (c)).

MDCK monolayers seem to exhibit very different creep behaviour when myosin II activity is reduced. Whilst untreated monolayers reach a steady strain value after about 100 s, Y-27632 treated tissues continue to flow in a power-law manner (see figure S5), suggesting a qualitative difference between the two systems. This apparent contrast is however properly accounted for by the model. The predicted creep responses, calculated using the parameters obtained from figure 2, are in good agreement with the experimental data with no free parameters. Based on the parameters obtained form relaxation experiments on Y-27632 treated monolayers, *τ*_{1} and *τ*_{2} are both much larger than in the untreated case, and now comparable in value with the duration of the measurement (see table S2 in Supplementary materials). Furthermore, the reduction of *ξ* in the treated case, down to 0.7 compared to about 1 in the untreated case (figure S5 (c)), would change the shape of creep curve and give the impression that creep accelerates rather than saturates at times close to *τ*_{1}(figure 3(b)). This analysis illustrates how modelling can bring consistency across systems that may at first appear qualitatively different, in particular when observed over a finite experimental time.

## Usage of the model beyond epithelial monolayers

Building on the work on MDCK monolayers, we may now consistently analyze data across biological systems, and pull information from different research groups, working with different mechanical testing protocols. For instance, we can show that the model successfully captures the relaxation response of single isolated cells, such as epithelial MDCK cells (figure 4 (a) [20], details on experimental setup in section 6) and articular chondrocytes (figure 4 (b), original data presented by Darling et. al. [35]).

Looking at sub-cellular components, the main factors controlling the mechanical properties of cells and tissues have been characterized experimentally already. Fischer-Friedrich *et al*. [7] have recently performed oscillatory compressions of HeLa Kyoto cells during mitosis by using an AFM cantilever and analysed the data to extract the rheological behaviour of the cell’s cortical actin network. To allow the direct comparison between our constitutive model and the rheological data introduced by Fischer-Friedrich *et al*. [7], we calculated the analytical expression of the complex modulus *G*′ (*ω*)+ i*G*″(*ω*) associated with our model (see section 7 in Supplementary materials) and fitted the experimental data again with an excellent agreement as shown in figure 4 (c).

Likewise, with the modelling framework presented here we have been able to capture the relaxation response of collagen fibrils (figure 4 (d)-top). Furthermore, using the parameters extracted from fitting the relaxation data, we have been able to predict their creep behaviour (figure 4 (d)-bottom); a quantitative link that was absent in the original paper presented by Shen *et al*. [36].

As many biomaterials exhibit power law rheology, examples where the generalized fractional viscoelastic model successfully captures their behaviour abound in literature–e.g. blastomere cytoplasm and yolk cell rheology (see figure S12 (c)-(d) in Supplementary materials). Studies available in the literature have reported simpler qualitative creep and relaxation behaviour for single cells, such as a single power-law or a two-power-law behaviour [37]. These responses are embodied as special cases of the presented generalized model (negligible stiffness and/or large viscosity). To illustrate this, we fitted the power-law response of single immune cells (figure S13). We could also capture the creep response of a single muscle cell exhibiting a two-power-law behaviour, and predict its storage and loss modulus (figure S12 (a)-(b)). This sets the fractional viscoelastic model presented here as a promising tool to support a unified description of the mechanical response of a broad variety of biological tissues.

## Links with biophysical analysis

Fractional models capture with a few parameters complex power law behaviours commonly associated with a broad distribution of relaxation times. A single spring-pot element is for instance sufficient to model a power law response, with two parameters. The complexity and richness of the spring-pot is apparent in the way the fractional derivatives are calculated, requiring an integration over the whole deformation history of the material, whereas the response of traditional elements only depend on the instantaneous values or rate of change of the strain. This integral expression of the derivative is responsible for the strong history dependent effects associated with power law rheology. The approach becomes even more powerful when combining the spring-pot element with other rheological components. Based on dimensional analysis, we identified two characteristic times and an effective stiffness involved in master curves for both the relaxation and creep functions. In the case of creep, the ratio of the two time-scales controls the qualitative shape of the response, accounting for seemingly different responses of the material with and without contractility. This provides ways to bridge our mathematical framework with more physical descriptions of the rheology of cells and tissues. For example, Fischer-Friedrich *et al*. [7], after observing that a simple power law function failed to capture the dynamic response of HeLa Kyoto cells, built a physical model based on a flat distribution of relaxation times up to a certain cut-off time. The model we present here captures a similar phenomenology by introducing a dashpot in series with a springpot. The characteristic time *τ*_{1}defined for the fractional model acts as a cut-off in the relaxation spectrum, causing a transition from a power-law to an exponential behaviour in the relaxation function. In both studies, a reduction of myosin activity leads to an increase of the characteristic or cut-off time scale. The model proved to be useful to compare two different biological systems, Hela and MDCK, and identify similar behaviours despite the use of distinct testing approaches.

More generally, having a unified language to consistently capture the response of materials over a broad range of time scales is a significant step towards the understanding of the physical and biological significance of rheological behaviours, one of the current key challenges in Mechanobiology [38]. For instance, by comparing the relaxation response of single MDCK cells with MDCK monolayers, we noticed that they display similar behaviours (figure 4 (a)). However, looking at the parameter values (see table S2) reveals that monolayers exhibit a higher stiffness *k*, “firmness” *c*_{β,} and viscosity *η*. Quantifying these mechanical properties raises novel questions, and we can only speculate at this stage about the reasons. The increment in stiffness *k* displayed by the suspended monolayers compared to single cells may for instance be ascribed to an increase in cortical contractility as previously reported [20]. By contrast, the origin of the higher “firmness” of the spring-pot and viscosity *η* could be related to changes in the internal cell organization when cells form intercellular junctions to integrate into a monolayer. In the context of single cells, the relaxation plateau observed when cells are deformed could be associated with cortical tension [7], confirming further a strong link between *k* and cortical activity.

Although cells often exhibit non-linear behaviours, the model provides a rich base line from which to assess unusual traits of their response. Recently, Bonakdar *et al*. [21] performed local measurements of the rheology of mouse embryonic fibroblasts by imposing cycles of loading at constant force and relaxation using a magnetic particle linked to the cytoskeleton through the cell membrane by a fibronectin coating (figure 5 (a)). They observed a power law response coupled with an incomplete cell recovery suggesting plastic deformations that tend to decrease in amplitude with the number of loading cycles. Such a local mode of deformation is difficult to model quantitatively due to the complex and undetermined interaction between the bead and cytoskeleton. We nonetheless simulated such cycles of loading for an arbitrary amplitude of imposed stress, using the model parameters previously obtained from the MDCK cells. At first, we only considered a simple power-law behaviour (i.e. a single spring-pot) and confirmed the results of Bonakdar *et al*. that a power-law alone cannot reproduce the progressive drift of the bead displacement or cell deformation observed experimentally (figure 5 (b)). We then tested the full model obtained for MDCK cells; here again, the model provides deformation that quickly return to 0, missing the deformation drift visible in the experimental data (figure 5 (c)). However, as discussed earlier, the spring *k* appears to be primarily linked to the overall deformation of the cell cortex, providing a restoring force if the cell is squashed or elongated. Given the local nature of Bonakdar *et al*.’s measurements, we expect that the elastic term *k* originating from cortical tension is likely to be irrelevant as a first approximation. By removing accordingly the contribution of the spring, the response to cycles of loading does provide a drift that is more consistent with experimental data, albeit growing linearly with time and exceeding the experimental trend (figure 5 (d)). However, it has been shown previously that during application of a local stress through fibronectin coated beads, cells respond with a local strengthening of the cytoskeleton linkages [39]. By reinstating a small proportion of the spring value to account for such local stiffening we could generate a series of curves consistent with the experimental data over the first 80 seconds. Although this analysis may not fully account for the non-linear plastic behaviour reported in mouse embryonic fibroblasts, it illustrates very clearly that a number of qualitative distinctive traits, reported in different systems and under different experimental conditions, may be in fact largely consistent with each other.

## Conclusions

We have presented a constitutive model for epithelial monolayers that captures the biphasic nature of their stress relaxation dynamics over their full physiological functional range. From a qualitative analysis of the monolayers’ relaxation response, we determined the number of parameters required to account for the behaviour observed, and combined traditional viscoelastic elements with the fractional spring-pot to fit tissue responses. The particular rheological model introduced here has been validated for the first time against experimental data for the relaxation response of suspended MDCK monolayers. Strikingly, the values for the parameters obtained by fitting the relaxation response could predict the monolayer’s response to other mechanical tests with excellent accuracy. This confirms that our model provides a constitutive description of the material and that the fitting parameters are proper material properties independent of the type of deformation or force applied to the material. We further demonstrated the model’s suitability to analyse and compare rheological behaviour across many systems and length scales, within the same unifying framework. It greatly facilitates a biophysical analysis by enabling a more intuitive approach to power-law modelling, and allowing us to ask more direct questions regarding the biological significance of the parameters involved.

Beyond enhancing our understanding of biological systems, a unified rheological model for biomaterials is also crucial in the medical and engineering fields. Correlating changes in the mechanical response of tissues to their biological state has been long considered as a promising marker-free method for cancer diagnosis [40]–[42]. In the context of regenerative medicine, rheological phenotypes provide a suitable metric to assess the similarity between tissue engineered constructs produced *in vitro* and their natural counterparts. As the parameters of the model are material properties that are independent of the method of measurement, they are promising mechanical signatures of the tissue and its condition to be used for diagnosis or as a target for regenerative medicine.

A practical limiting factor for the widespread application of the model presented here is the mathematical complexity of fractional derivatives, and the current lack of user-friendly numerical methods to perform such analysis without expertise in fractional calculus. Such tools have been recently released in the public domain by our group [43]. This will ensure a broader adoption of fractional models for a rapid and systematic analysis of experimental data, as well as future integration within numerical packages and finite element software.

## Author contributions

A.B and A.K. designed the rheological model. N.K carried out the relaxation and ramp experiments. J.F. and G.C designed the creep experiments. J.F. carried out the creep experiments. A.B and A.K. performed data analysis. J.F. and G.C. contributed to physical interpretation of the data. All authors discussed the results and manuscript.

## Acknowledgements

The authors wish to acknowledge present and past members of the Kabla and Charras labs for stimulating discussions. A.B. and A.K. wish to acknowledge Louis Kaplan for his important contribution to the development of the RHEOS package and Arran Fernandez for stimulating discussions on fractional calculus. A.B and J.F. were funded by BBSRC grant (BB/M003280 and BB/M002578) to G.C. and A.K. N.K. was part of the EPSRC funded doctoral training program CoMPLEX. N.K. was also funded by the Rosetrees Trust, and the UCL Graduate School. G.C. is supported by a consolidator grant from the European Research Council (MolCellTissMech, agreement 647186). The work was supported by BBSRC grants (BB/K018175/1, BB/M003280 and BB/M002578).