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We propose a mathematical model to suggest a unified explanation behind the observation that some cell types tend to spread more efficiently on stiff substrates and are able to adapt their internal stiffness to the external stiffness. Our model also offers an explanation regarding the dependence of cell spreading on cell type. We show that our model for stiffness adaptation is in good agreement with experimental data. We also apply our model to calculate the energy of traction on bulk substrates as well as thin coatings, thereby extracting estimates of critical coating thickness as a function of cell type and coating bulk modulus.

Recent work suggests that the mechanical properties of a cell’s microenvironment can have as great an impact on cell structure and function as soluble stimuli and cell-cell contacts [1-11]. The intracellular and extracellular responses to mechanical forces depend both on the material properties of the cell itself and the properties of the surface or matrix to which it is attached [12-15].

While the responses are specific to cell type [3,15,16], there is a clear dependence of cell behavior on substrate rigidity. Specifically, cellular shape, size, and extent of spreading are determined by a specific range of rigidity values that depend on the tissue type from which the cells are derived. For example, certain types of cells grown on stiﬀ substrates exhibit a more spread phenoltype [

As cells alter their morphology and downstream behavior when grown on substrates with different rigidity, it is natural to expect that the internal cytoskeletal assembly will change accordingly.

Correspondingly, the internal stresses and the cell stiffness might be expected to change in response to the rigidity of the substrate. Experiments performed on fibroblasts cultured on fibronectin-coated PA gels of varying rigidity [25,28] and alveolar macrophages cultured on collagen-coated PA gels of varying rigidity [

Finally, Evans et al. [

Taken together, these studies show that matrix stiffness plays a key role in influencing cell morphology and, crucially, downstream stem cell behavior.

Several models have been proposed to explain and describe the influence of substrate stiffness and thickness on the mechanobiology of cells. Building on the classic work by Eshelby [

However, what has been lacking thus far is a simple model that links the energetics of cell traction on bulk substrates with the influence of substrate thickness on cell traction and that can thus guide the optimal design of substrates to enable efficient cell traction.

In this article, we describe a model that is primarily concerned with the energetics of cell traction and that focuses on three distinct but related observations concerning the connection between cell behavior and the stiffness of the substrate/coating on which cells are cultured:

1) Stiffer substrates encourage more efficient and increased spreading of some cell types [1,2,7,9,16,22-27].

2) As part of the response to the mechanical properties of the substrates, some cell types, e.g. fibroblasts, are able to adapt their internal stiffness to substrate stiffness better than other cell types [25,28].

3) Some cell types (e.g. fibroblasts) spread more on soft, thin gels [2,23,25,44,45] and thus show that they are able to sense the effective stiffness of a thin soft gel attached to much stiffer supports, e.g. glass.

In contrast to earlier detailed models [33-38], our model is primarily concerned with the energy involved in the traction process and hence does not describe cell shape or cell spreading. Our model takes into account the interaction between internal cell stiffness and external stiffness in a simple fashion as described by Schwarz et al. [13,14]. Hence, we do not describe either the details of focal adhesions as have been done in [12,36-38] or the role of biochemical coupling to cell mechanics as described in [39-41]. Crucially, however, our model recognizes the fact that the substrate stiffness perceived by the cell can be thickness-dependent and incorporates the models of Maloney et al. [

We further note that cell-material interaction is mediated by multiple focal adhesions in reality and the interaction is a very complex process. Therefore, as a first step, we focus in this paper on describing the energetics of a single focal adhesion complex interacting with a substrate. Hence, throughout this paper, when we talk about cellular traction, it is in the context of the traction associated with a single focal adhesion interacting with a substrate.

The model in this paper is chiefly concerned with the energetics of cell traction. For the purposes of this paper, we will define traction to be the process by which a cell experiences forces exerted by the underlying substrate through focal adhesions (FAs) and the resultant mechanical response of cells. As a cell experiences forces exerted by the substrate, for it to initiate a mechanical response to these forces, the cell has to invest a certain amount of energy to trigger a complex set of downstream responses that include but are not limited to adhesion, spreading, and motility. Therefore, understanding the energetics of the traction process could allow us to explain differences in downstream cell responses.

Specifically, in designing coatings on surfaces for adherent cells to demonstrate certain functional behavior, the ability to predict differences in cellular traction for different combinations of cell types and underlying coating/substrate stiffness and thickness is very valuable. For example, this understanding can help in answering a specific question such as: when cells of different types are cultured on thin coatings of variable bulk stiffness fixed to stiff substrates, what is the impact of the thickness and bulk stiffness of the coating on the energetics of the traction process in particular? Using our model, we will tackle this question in this paper.

Our model builds upon two existing but separate models: 1) one due to Schwarz et al. [13,14] for substrate rigidity sensing, and 2) the other due to Maloney et al. [

Schwarz et al. [

the focal adhesion takes place through a spring whose spring constant is denoted by . This spring constant represents the extracellular matrix (ECM) elasticity. Later in this article, we will invoke the treatment by Maloney et al. [

Since the internal spring and external spring are in series, the effective spring constant of the system is given by

In the original two-spring model of Schwarz et al. [

where is the free velocity of the motor and is the stall force. Typically the free velocity of the motor is of the order of and the stall force is of the order of few pN [

As the cell strains, the molecular motors invest power which is derived from the power stored in the springs. Thus, we have

where we have used Equation (2). We can integrate Equation (3) trivially to yield the dynamics of force build-up over time,

where

Thus, larger the effective spring constant , smaller the time constant , and more efficient the force build-up. We can compute the total energy invested by the cell in the traction process in a straightforward manner,

We wish to make it clear that although the force generation in cells in response to external influences is dynamic in nature [13,47-49], our paper will not deal with dynamic, time-dependent traction processes.

In order to make further progress, we need a connection between the external elastic modulus , and the external spring constant , of the substrate as perceived by the cellular focal adhesion. Fortunately, the model by Maloney et al. [^{1}

Throughout this article, we will set Poisson’s ratio, to 0.5, and assume a focal adhesion size of consistent with the values used in [

As observed by [2,23,24,44,45] among others, when some cells are cultured on soft, thin substrates that are attached to stiff supports, these cells are able to “feel” the effective stiffness of the composite substratum. Thus cells are able to spread far greater than what the soft substrates alone would suggest. Maloney et al. [

where is a thickness dependent normalized function that can be approximated very well for our purposes as [

.(9)

We have thus far described earlier models due to Schwarz et al. [13,14] for substrate thickness sensing. We will now discuss how we extend these models to include the phenomenon of stiffness adaptation of cells. As described by Solon et al. [

In order to compare our proposed heuristic model with experiment, we rewrite Equation (10) in terms of elastic modulus as:

We display in

(or equivalently ) gives the best fit between the model and experiment.

In

) gives the best fit between the model and experiment.

Equation (10) (or its analogous Equation (11)) naturally gives us two limits: for , and for . Equation (10) (or its analogous Equation (11)) also suggests that cell types with larger have greater intrinsic ability to gain traction by greater lowering of traction energy. This suggestion is supported by experiments [16,25,28] that show that some cell types are more able than other cell types to modify their internal stiffness and thereby spread more easily on stiff substrates.

We note here that our heuristic model for cell adaptation presented in Equations (10) and (11) is by no means unique. Other possible models could be chosen with similar limiting forms as the one that our model gives. We have chosen our model so that it is simple enough and yet give the right adaptation behavior. Clearly, a combination of more detailed experiments and models are required to give a more accurate adaptation model. We further recognize that stiffness adaptation involves a complex interplay between two distinct aspects of the mechanochemical transduction process of cell surface receptors: 1) recognition of differences in substrate rigidity and 2) initiation of intracellular changes resulting in cytoskeletal reorganization. Combining Equations (1), (6) and (10), we obtain the total system energy invested by the cell during the traction process:

Since we are primarily interested in changes in traction energy as a function of substrate properties, we normalize the expression given in Equation (12) with respect to its limiting value at small external stiffness,

Throughout the paper, we will focus on the normalized value of the traction energy as shown in Equation (14). As has been noted earlier in the paper, whereas the force generated by cells increases monotonically and dynamically with substrate rigidity [47-49], we focus in this paper on traction processes involving single focal adhesion complexes. The increase in force generation within cells is likely caused by the recruitment of more proteins and hence an increase in number of focal adhesion complexes.

The expression for traction energy in Equation (12) immediately suggests that systems with larger effective spring constants will lead cells to invest less energy in the traction process and can thus help cells gain traction more easily. This suggestion in turn implies the following:

1) Substrates those are stiffer, either because they have intrinsically larger elastic modulus or because they are thin and attached to stiff supports, will yield larger effective and thus result in more efficient traction process. This is in line with experimental observations [1,2, 7,9,16,22-27].

2) It is energetically favorable for cells to raise their internal stiffness to match the stiffness of the ECM as is observed for ﬁbroblasts by Solon et al. [

We are now in a position to discuss the energetics of the traction process when cells of different types spread on substrates with a range of elastic moduli. In

As is to be expected, the curves in

As observed in [2,23,25,44,45] among others, when some cells are cultured on soft, thin substrates that are attached to stiﬀ supports, these cells are able to “feel” the eﬀective stiﬀness of the composite substratum. Thus cells are able to spread far greater than what the soft sub-

strates alone would suggest. We will now apply the methods of the earlier Sections 2.1, 2.2, 2.3, and 3 to discuss the energetics of the traction process when cells of different types spread on substrates with fixed or variable intrinsic elastic moduli but different thicknesses. Combining Equations (7)-(12), we can write the energetics of the traction process for cells whose focal adhesion size is given by on coatings of thickness denoted by h as:

Again, as before, we are primarily interested in the traction energy normalized with respect to its value at bulk substrates, as given by:

We can now analyze the dependence of the energetics of the traction process on cell type as well as thickness of coating on which the cells are cultured by studying

As the figure shows, thin coatings substantially lower the energy required for the traction process. However, this lowering of energy can be truly beneficial only for cells that have the necessary internal stiffness and cytoskeletal machinery, i.e. sufficient . For instance, whereas the traction energy is lowered two-fold for cells with for thin substrates compared to their bulk counterpart, stiffer cells with

are able to lower their traction energy by more than a factor of 20.

We can further investigate the reduction of traction energy for thin substrates by studying the reciprocal of the normalized traction energy given by Equation (16) evaluated at a substrate thickness, h and denote this quantity by . We evaluate at a thickness of h = 0.01 μm and display it as a function of cell stiffness and various values of bulk substrate stiffness in

Since the thickness of soft coatings can significantly modulate traction energy, the question of an optimal or critical thickness for a given cell type and function assumes immediate importance for several conceptual as well as practical reasons. First, while the chemistry of the coating can be tuned [

or critical thickness stems from the consideration that using substrate coatings that are more than 1 μm thick may be undesirable from a processing and imaging point of view. One would thus desire a coating that would be as thin as possible and yet be able to effectively modulate cell function and phenotype.

In the context of the present paper, for the sake of simplicity, we will characterize the energy of traction to be a cell phenotype that can be modulated by coating stiffness and thickness. Further, we will monitor the ratio of the traction energy the cell has to invest for a bulk coating to the traction energy invested by the cell when cultured on a coating of thickness h. As _{crit} beyond which we can state with some confidence that the coating has ceased to modulate cell function; instead cell function is dictated predominantly by the elastic modulus of the stiff substrate on which the coating is fixed. We plot in _{crit} as a function of intrinsic cell stiffness k_{i-max} (i.e., different cell type), where we have fixed the threshold value to be equal to 2. In

for cells in the stiffness range of 2 - 10 pN/nm, coatings with a bulk modulus of E_{bulk} = 1 kPa need to be in the thickness range of 0.35 - 0.5 μm so that the cells can lower their traction energy by at least a factor of 2 compared to the cells cultured on bulk coatings.

We have proposed a model that is chiefly concerned with the energetics of cell traction with a focus on the observations that some cell types tend to 1) spread more efficiently on effectively stiff substrates and 2) better adapt their internal stiffness in relation to the external stiffness of substrates. We have built on ideas described in a simple two-spring model for rigidity sensing by [13,14] and a model for substrate thickness sensing by Maloney et al. [

1) Using a stiffness matching model we estimate, the maximum cell stiffness for fibroblasts and hMSCs when cultured on polyacrylamide coated glass to be 5.5 kPa and 6.8 kPa.

2) Stiffer cells that can raise internal stiffness by saying a factor of 20 with respect to softer cells will require almost 20 fold lesser energy for the traction.

3) Finally, we provide a preliminary phase diagram of coating thickness needed for cells of different internal stiffness to require a change of no more than a factor of 2 in energy invested in the traction process.