Cells adapt their shapes with their experienced stiffness: A geometrical approach to differentiation

Journal of Theoretical Biology 486 (2020) 110086

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Cells adapt their shapes with their experienced stiffness: A geometrical approach to differentiation Amir Hossein Haji Department of Materials Science and Engineering, Shiraz University, Shiraz, Iran

a r t i c l e

i n f o

Article history: Received 11 September 2019 Accepted 17 November 2019 Available online 18 November 2019 Keywords: Stiffness-directed differentiation Cerruti solution Cell internal-external adaptation Intracellular distance

a b s t r a c t The elasticity-directed differentiation of mesenchymal stem cells has been widely studied since mid 20 0 0s. Over nearly linear-elastic materials the differentiation of the stem cells are shown to be related to the Young’s modulus of the substrate. While it is found that constraining the stem cells to some prepatterned shape and size affects their differentiation, it has been also well recognized that the cell morphology intimately relates to its type and functionality such that the cell morphology has been accepted as an indicator for the differentiation path. This paper conjectures the importance of geometry in differentiation and claims that the elasticity indicator for differentiation is indeed "the stiffness" which contains not only the elasticity coefficient but also the geometrical information. An elasticity model is derived for a singular cell with different shapes over a thick substrate, which almost resembles the condition for most of the tests with sparse distribution of the cells. Analysis of principle shapes such as square, rectangle, and hexagon (with and without dendrites) suggest that the larger the aspect ratio i.e. the further the shape to the roundness, the larger the substrate stiffness experienced by the cells. By moving towards a more round shape such as a hexagon with or without dendrites the substrate stiffness falls off rapidly. Then by including the stiffness of the cell body itself we arrive at a more important finding; the cells at free culture condition prefer the shape which best equalizes their experienced stiffness of the substrate and that of their own body. The body-to-substrate stiffness ratio, hence, explains why a slender rectangle is the preferred shape at myogenic differentiation range and a hexagon with dendrites is the preferred one at neurogenic range. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Surveys on effective physical factors in morphogenesis are vastly developed in recent years. Stimuli come from the fact that physical and biochemical activities of the cells take place concurrently while certain interactions can transform their signals into common forms. Physical activities of the cells proceed in the context of a complex interlaced structure with strong coupling between the internal and external dynamics of each individual cell as well as the interactions of the cell with its surroundings (matrix and other cells). As due to their indisputable role in development and health, as well as enhanced controllability/stability compared to biochemical manipulations, physical mechanisms of cellular scale invoke research challenges towards fascinating tissue engineering applications. The changes in cell shape, contraction (and/or adhesive) force, cell deformation and spatial relocation, for instance, are intimately related to the mechanics of the cell and its surrounding. Nu-

E-mail address: [email protected] https://doi.org/10.1016/j.jtbi.2019.110086 0022-5193/© 2019 Elsevier Ltd. All rights reserved.

merous evidences are recently found for mechanical factors interfering the formation of tissues (Li et al., 2009; Nelson, 2009; Moore et al., 2005, Engler et al., 2006). The significant feature relevant to more or less all such observations is that the cells continuously generate and apply force particularly to actively sense the properties of their surroundings. For rigidity sensing, for example, cells must actively strain their surroundings to probe the elastic properties. The generation of the contractile force requires changes at subcellular scale (rearrangement of actin cytoskeleton and myosin motors (Schwarz and Safran, 2013)) while it also provides feedback (deflection) for even more internal changes. Rather an interesting relationship indeed is found between the ECM stiffness and the cell lineage specification. Mesenchymal stem cells (MSC) cultured on relatively soft polymer gels which mimic the ECM elasticity of brain, are differentiated to neurons. MSCs on a stiffer matrix similar to that of a muscular tissue are differentiated to muscles, and those on a comparatively rigid matrix that mimics collagenous bone are differentiated to bone (Engler et al., 2006, Reilly and Engler, 2010, Ghosh and Ingber, 2007). A comprehensive collection of results including neurogenic, adipogenic,

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A.H. Haji / Journal of Theoretical Biology 486 (2020) 110086

myogenic, cardimyogenic, chondrogenic, and osteogenic differentiations at various mechanical-chemical conditions are summarized in Lv et al. (2017). The Young’s modulus criterion for elasticity-directed differentiation, however, cannot satisfactorily explains various results over more realistic nonlinear materials (with different elasticity/density conditions). For instance, two soft substrates with Young’s moduli in neurogenic range (0.1 kPa < E < 0.5 kPa) trigger different cell density distributions and correspondingly different differentiations (Saha et al., 2008). In uniform distributions of the cells more neurogenic differentiation is observed compared to non-uniform distributions (variable-density domains) which cause many cell clusters (Saha et al., 2008). This is an example of quite a common observation in experiments (simultaneous differentiation to a variety of cell types on the same substrate or substrates with the same modulus) which implies that the elasticity indicator needs to be a more representative one for the complicated cell-surroundings interaction and particularly should contain some geometrical information too. The stiffness-directed differentiation is therefore the preferable version compared to the original one which was based on the Young’ modulus alone. Due to variety of the nonlinear materials and their complicated stiffness hardening relations, however, it is necessary to have the stiffness computation (experienced stiffness by the cells) on a linear material basis. Actually the tests are also done usually on such materials (almost linear material) to collect a database. So an analytic approach is necessary to estimate the experienced stiffness of the stem cell over a linear material. Here I establish an elasticity framework for a singular cell of different shapes over a thick substrate. The experimental stiffness can be normalized by such stiffness at thick linearly-elastic substrate mode to establish some differentiation criteria. Via such criteria differentiation type of the cells over various linear/nonlinear materials can be inferred by just resorting to the available experimental results of differentiation of the cells at elastic thick substrate mode. From another point of view, however, I aim to show even some more significant evidences for the stiffness role in differentiation. It is well recognized that the cell shape has a strong relation with its type and functionality and actually plays as a major indicator in determining the differentiation path. Also constraining the stem cells to some pre-patterned shape and size are found effective in controlling their differentiation (McBeath et al., 2004; Kilian et al., 2010). I argue that the dependence of the cell’s experienced stiffness (and hence, its differentiation) to its shape and size analytically corroborates such findings. It is also found that the changes in the cell cytoskeleton properties and particularly the Young’ modulus tend to comply with the elasticity of the substrate (Solon et al., 2007). I discuss such activity is beyond just some local adaptation to the substrate properties. Indeed the stiffness-based approach reveals that there should be an adaptation at whole cellular scale such that the cell changes its shape (along with its Young’s modulus) to optimally match the stiffness of its own body and that of the substrate. 2. The stiffness experienced by a singular cell Analytical determination of the stiffness experienced by the cells in different conditions has been developed in recent literature (Deshpande et al., 2008; Friedrich and Safran, 2012; He et al., 2014; Salbreux and Jülicher, 2017). Such models can be adequately employed to address questions in substrate mechanosensing by a single cell, however, the question about the stiffness dependence to the shape as well as configuration is not explicitly answered. In what follows some linear elasticity solution corresponding to a tangential force applied on a thick substrate (the Cerruti problem)

Fig. 1. A singular cell under the reaction forces by the substrate (a), and the traction forces applied by the cell to the substrate (b).

is utilized for a square cell. The Cerruti solution in its common form relates a point (singular) force to its generated displacements on a half-space (Sadd, 2005):

U (x, y ) =

V (x, y ) = R=

P



4 π μR



1+

x2 + ( 1 − 2ν ) R2

1−

x2 R2



(1)

P ν xy 2 π μR 3



x2 + y2

(2)

where μ is the shear modulus of elasticity and related to the Young’s modulus and Poisson’s ratio by μ = E/(2(1 + ν )), and U and V are the displacements at (x, y, 0) due to a tangential point force Pexerted on a half space at (0, 0, 0) and directed along xaxis. We omit the z component hereafter as the traction forces are exerted at the surface and the displacements are also needed to be only calculated at some surface location. The Cerruti model is indeed a proper choice to approach a contractile cell in most of the experiments where the cells are cultured at low densities over a substrate much thicker compared to their dimension. However, as a contractile cell exerts some distributed force over its edges, we need an extension of the common Cerruti solution to the distributed-force version. For the simplest case we take a square cell of size l (side length equal to l) to set up a framework which will be systematically extended to various geometries later. 2.1. Square cell Consistent with a 2-dimensional Cartesian framework the square shape is the simplest case which would serve as a basis for comparison between different more realistic cell shapes in the next sections. Assume a square cell applies along its sides some uniform traction forces to a half space (the substrate) as shown in Fig. 1. The tractions are generally believed to reach their maximum at the cell periphery while diminishing rapidly towards the center of the cell (Schwarz and Safran, 2013; He et al., 2014). As an approximation to this, the tractions here are taken uniform over narrow bands of size l × δ on each side as shown in Fig. 2. The generated displacement by the cell at any spatial point would be obtained considering the effect of all the infinitesimal

A.H. Haji / Journal of Theoretical Biology 486 (2020) 110086

3

where f2 ((l∓δ )/2; 0, l) denotes the integration of the Eq. (2) over the x band: (l + δ )/2 to (l − δ )/2 and y band: 0 to l. Carrying out the integration gives (see Appendix A):

uS3 =

Fig. 2. Distributed surface tractions over the left side of the square of Fig. 1 (left) and the coordinate system directions to compute displacements (right).

traction forces distributed over the 4 sidebands. Consider, for instance, the middle of the left side of the cell; point A. We may attach the coordinate to this point and pursue its displacement in x direction i.e. the x-displacement at (0,0). The contribution of the traction band of the right side of the cell to the x-displacement at (0,0) can be obtained by integrating the Cerruti’s solution given in Eq. (1):

uS1 = −2 f1 (l ± δ /2; 0, l/2) = 2 

l/2

−2 p˜



l+δ /2 l−δ /2

(1 − ν ) +



4π μ x2 + y2

0



ν x2

 dydx

x2 + y2

(3)

p˜ = P/(l × δ ) and P being the distributed and total traction forces respectively and f1 (l ± δ /2; 0, l/2) denoting the integration of the Eq. (1) over the x band: l − δ /2 to l + δ /2 and y half-band: 0 to l/2. Carrying out the integral yields (see Appendix A):

    2  l+δ/2  −(1 − ν ) p˜x  l l  uS1 = ln  1 + + πμ 2x 2x   l−δ /2 l+δ/2   2x 2 2x  p˜l   + l n 1 + −  2π μ  l l 

(4)

l−δ /2

The displacement at (0, 0) due to the tractions over the left side can be calculated in a similar way:

uS2 = 4 f1 (0, δ /2; 0, l/2) = 4 

l/2 0

2 p˜



4π μ x2 + y2





+δ /2 0

(1 − ν ) +

ν x2



x2 + y2

dydx

(5)

which gives:

    2  +δ/2  l l  (1 − ν ) p˜x  uS2 = 2 ln  1 + + πμ 2x 2x   0 +δ/2   



2 p˜l 2x 2x   − ln 1+ −  π μ  l l 

(6)

0

Also Eq. (2) would be used to calculate the x-displacement at (0, 0) due to the distributed tractions on the up and down sides:

uS3 = 2 f2 ( (l ∓ δ )/2; 0, l ) = 2

 (l−δ )/2  (l+δ )/2

l 0

4π μ

2 p˜ν yx  x2

+

y2

3 dydx (7)

− p˜ν

πμ

⎛

δ+



p˜ν ⎝ l2 + l + δ πμ 2



2 −

 l2 +

l−δ 2

2

⎞ ⎠

(8)

uS1 , uS2 and uS3 sum up to the total displacement of the point A; uSA . The Poisson’s ratio for cell/substrate could be taken ν = 0.3 (Schwarz and Safran; 2013) while also a small δ /l (around 0.1, for instance) would be a reasonable approximation consistent to the peripheral concentration of the traction forces (Schwarz and Safran, 2013; He et al., 2014). For such parameter values we get uS1 = −0.047 p˜l/π μ, uS2 = 0.309 p˜l/π μ, uS3 = −0.016 p˜l/π μ and uSA = 0.246 p˜l/π μ . As uSB = −uSA the absolute x-deflection of the line segment connecting A and B equals Sx = |−2uSA |. We can assume that for small deformations almost all the points on left and right sides of the square have the same displacements as A and B. As far as this no-shape-change assumption (the square remains square) is true we can assign the deflection of the line segment ABto the whole square i.e. the whole substrate portion under the cell body (later we will resolve this assumption). We are now ready to define a stiffness dividing the force magnitude (the magnitude of any of the two opposite forces on left and right sides: F = p˜l δ ) by the absolute x-deflection of the square:

KSqu =

F

Sx

=

p˜l δ

Sx

(9)

which is the stiffness of the substrate measured by a square cell. There should be cleared, however, why we have included the contributions of not only the left and right sides but all the 4 sides to Sx . The notion of the stiffness entails the concept of a pair of forces and the deflection between these forces. In the common methods of calculating the stiffness in mechanical structures, we set an arbitrary force on the desired location and find the resulting displacement, noting the fact that the equilibrium condition for the whole system is guaranteed by the supports of the system (which actually provide the reaction force to the arbitrary exerted force). In a similar way the stiffness can be measured between two structural locations by applying opposite forces at these locations. This is the way a cell measures the stiffness of the substrate. If the cell remains in-place we might think it sufficient, as due to the symmetry, to only consider the effects of the left and right sides of the cell. However, from another point of view, the cell behaves as a whole and the four tractions (on all sides of the square) should be applied at the same time. So while it is actually not a necessary condition to consider all the forces to calculate the deflection, it is in fact a constraint applied by the system; the simultaneity of the cell tractions. For typical parameter values ν = 0.3, δ /l = 0.1 we get KSqu = 0.24El. It is noticeable that the stiffness is proportional to the cell size l which implies that by increasing the cell size a larger portion of the semi-infinite space resists the contractile force. The analytical result of the above model also well agrees with experimental results for collagen (Han et al., 2018) (see Appendix B). The protrusion length effect is demonstrated in Fig. 3 which shows by more deviating from a concentrated to a distributed force the stiffness is experienced harder. This indeed complies with the expectation that a more concentrated force generates more displacement. At the end of this section, it should be noted again that the analytical procedure developed here was for a square cell at sparse condition. However, the extension of this procedure to some population of the cells can also be easily pursued (see Appendix C). Such extension would be used for finding the substrate stiffness

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A.H. Haji / Journal of Theoretical Biology 486 (2020) 110086

Fig. 4. Degeneration of a square cell and formation of cusp corners (a), Distribution of the contractile nodes (b)

Fig. 3. Protrusion length effect on the experienced stiffness by a square cell.

experienced in a 2-dimensional cellular lattice and investigating the density effects. 2.1.1. Body stiffness of a square cell So far we discussed how stiff the substrate is measured by a square cell. Here we deliberate on the stiffness of the cell body itself. Generally the cell’s traction can be modeled by a tensioned spring (the entire cell body) attached to some springs at rest (the matrix) (He et al., 2014; Basu et al., 2018). If the cell stiffness is larger than the matrix stiffness KCell > KMatrix (where KMatrix depends on the cell shape as expressed before; for instance, KMatrix = KSqu for a square cell) the cell pre-elongation reduces i.e., the cell tends to its rest length while at the same time it contracts the matrix. On the other hand, if KCell < < KMatrix the cell deflection almost remains unchanged as it cannot compress the matrix. KCell ~ KMatrix , however, resembles the case of two similar springs in parallel and hence, the deflection (pre-elongation) of the cell reduces to almost one-half of its initial value. In order to find KCell for a thin square cell we assume a planestress situation i.e., the reactions from the substrate (Fig. 1(a)) are assumed 2D normal stress across the cell body. Furthermore as σxx = σyy , τxy = 0 the case is actually a hydrostatic (isotropic) one:

1 (σxx − νC σyy ) EC (1 − νC )σC l l = ε × l = EC P σC (l × z ) EC z KCel l (Squ ) = = = l l 1 − νC

εxx = εyy =

by the large-magnitude tractions as well as non-elastic transformation due to the subcellular rearrangement of actin cytoskeleton and myosin motors. This new deformed shape, however, serves as the effective (functional) shape by which the cell experiences its surrounding stiffness only via small variations (due to small variations of the traction force). Therefore for the new deformed shape we can make the no-shape-change assumption (small deformation) again. Assume, for instance, that the square sides concave (Fig. 4). The cusp corners of such degenerated square can be modeled as triangles attached to the original square where some parts of the sides are still straight. This model is actually a 4-sided version of the neurons star-shape model (with 6 sides) presented in Section 2.3. Also in Section 2.2 we discuss a symmetry breaking transformation of the square i.e. the case where the sides break the square symmetry in one direction and eventually form a slender muscle cell. 2.1.3. Discussion on cell size effect and reshaping, the concept of body-to-substrate stiffness ratio According to the equation set (10) for two similar cells of sizes l1 and l2 < l1 the larger cell experiences its own body stiffer since z1 = (l1 /l2 )z2 > z2 . This is somehow in contrast to a common naive belief that small objects always are harder than larger ones which most probably originates from the elastic rods model. For an elastic uniform rod of length L and cross sectional area equal to A, under an axial tension/compression force equal to F, we may write:



F = σ .A =

(10)

where EC is the cell’s Young’s modulus, σ C (= σxx = σyy ) the average stress at any cross section of the cell perpendicular to the line of action of the contractile forces and z the average thickness of the cell. 2.1.2. Discussion on no-shape-change assumption There is some crucial point about the no-shape-change assumption made for the square model which would be better resolved here before analyzing other cell shapes. By elementary mechanical knowledge we know that to assign some stiffness it does not matter how large the traction force is as far as the force-deflection relationship remains in the linear elasticity domain of the material behavior. However the magnitude of the tractions might be a matter of concern if the cell shape undergoes significant change. This is why we need to revise the no-shape-change assumption made for the square. In fact the square shape was just taken as a simple test case which would serve as a basis for comparison between different more realistic shapes. In such more realistic cases, the cell transforms into a new shape due to the elastic changes caused



EA L L

(11)

The stiffness K = EA/L relation clearly shows that by shortening the rod (the cross section remained constant) the rod becomes stiffer. However, if the rod is downscaled similarly in all dimensions it actually becomes softer simply because the area A(∝L2 ) decreases more rapidly than the length. This is the same behavior for two similar cells when one is a downscaled version of the other. So the matter of getting harder or softer when getting resized strictly depends on the type of the rescaling. For cells morphogenesis during differentiation, particularly at early stages it is a reasonable assumption that the cell transformation takes place under almost constant volume and constant attachment area. These assumptions give rise to the cell thickness (z) being almost constant. On the other hands, the similar rescaling (in 3D) is obviously impossible under constant volume assumption. However it is not only the constant-volume that rejects the similar rescaling. From the stiffness point of view, the cell reshapes to adopt its surroundings stiffness and this cannot be reached by just similar rescaling. Comparison between the stiffness of the cell body and the stiffness it feels from its surrounding can be achieved via defining the body-to-substrate stiffness ratio as:

RCM (Squ ) =

KCel l (Squ ) KMat (Squ )

(12)

A.H. Haji / Journal of Theoretical Biology 486 (2020) 110086

Fig. 5. Distributed surface tractions over the left side of a rectangular cell

Such ratio, indeed, is independent of the cell size as the cell body stiffness (KCell ) and the substrate experienced stiffness (KMat ) both have the same proportionality to the cell size. For square cell, for instance, as shown in Section 2.1 the substrate experienced stiffness by the cell KMat(Squ) for typical parameter values ν = 0.3, δ /l = 0.1 equals to 0.24EM l. On the other hand, equation set (10) gives the cell body stiffness equal to 1.42EC z. The body-tosubstrate stiffness ratio then reads RCM (Squ ) = 5.9 × (EC /EM ) × (z/l ). Under similar rescaling z/l remains constant and so does the bodyto-substrate stiffness ratio. By discarding the similar rescaling actually the natural selection for the cell shape change is reshaping. At 2D free culture condition i.e. when the cell freely can change its shape (in contrast to the pre-patterned culture condition) the reshaping would be volume and area conserved. In what follows we discuss reshaping effect corresponding to symmetry breaking in one direction and hence, formation of a rectangle. Then we do the same for a hexagon and hexagon with dendrites.

5

Fig. 6. Aspect ratio and protrusion length effects on the rectangular cell’s experienced stiffness relative to that of the square cell

creasing the protrusion (δ ), the force turns more concentrated, the deflection increases and hence, the stiffness lessens. 2.2.1. Body-to-substrate stiffness ratio for a rectangular cell For a rectangle the body stiffness should be found in both x and y directions. For the lateral direction (x), the equation set (10) turns to be:

w = εxx × w = P KCel l (Rctx ) = = w

(1 − νC )σC w EC

σC (L × z ) L EC z = w w 1 − νC

(14)

Similarly for the longitudinal direction (y):

L = εyy × L = P KCel l (Rcty ) = = L

(1 − νC )σC L EC

σC (w × z ) w EC z = L L 1 − νC

(15)

2.2. Rectangular cell

So the weighted average body stiffness of a rectangular cell KCel l (Rct ) = L × KCel l (Rctx ) /(L + w ) + w × KCel l (Rcty ) /(L + w ) reads:

For a rectangle of size w × L and traction bandwidth of δ (Fig. 5) we have:

KCel l (Rct ) =

uR1 = −2 f1 (w ± δ /2; 0, L/2 ) uR2 = 4 f1 (0, δ /2; 0, L/2 ) uR3 = 2 f2 ((L ∓ δ )/2; 0, w )

(13)

where in a similar way to the square, uR1 and uR2 are the contributions of the traction bands of the right and left sides (lengths) and uR3 is the contribution of the traction bands on the up and down sides (widths) of the cell to the x-displacement at (0, 0) (the middle point on the left side). The x-deflection is defined as Rx = |−2(uR1 + uR2 + uR3 )| which gives the x-stiffness as KRctx = F /Rx = p˜l δ /Rx . For the y-stiffness we may calculate the y-displacement replacing w and L in equation set (13). The total stiffness then is given by a weighted average of the x- and y- stiffnesses as: KRct = L × KRctx /(L + w ) + w × .KRcty /(L + w ). Under constant volume and area we pursue the stiffness changes corresponding to a square-to-rectangle transformation. As a typical case take the rectangle with aspect ratio 4: 1. Compared to the square dimensions l × l the rectangle reads w × L ≡ l/2 × 2l. With the same protrusion lengths as for the square cell i.e. δ /2l = 0.05 we get the stiffness of such rectangle equal to KRct = 1.88KSqu . Fig. 6 shows KRct /KSqu vs. inverse aspect ratio (w/L) for three different  values of protrusion lengths (the curves are plotted for w = l w/L ≥ 2δ ⇒ (w/L ) ≥ 4(δ /l )2 (. Remarkable is that for a typical muscle cell with w/L ≈ 0.05 the experienced stiffness can be more than 6 times the stiffness experienced by a square cell of the same area. The role of the protrusion length is as before, i.e. by de-

S + 1/S E z C 1 + 1/S 1 − νC

(16)

where S = L/w is the aspect ratio or spindle factor. The assumptions of the cell’s constant volume and constant attachment area  require that w = l w/L, L = l L/w and zRct = zSqu so:

KCel l (Rct ) =

S + 1/S

1 + 1/S

KCel l (Squ )

(17)

So by increasing the aspect ratio the stiffness of the cell body grows up very fast and indeed faster than the experienced stiffness of the substrate. Such faster growing rate is a compensation for the lower Young’s modulus of the cell compared to the matrix (EC < EM ). For example in myogenic differentiation range i.e. for EM ≈ 10kPa, the cell Young’s modulus just increases to EC ≈ 4kPa ( Solon et al., 2007). The cell thickness might typically be around 0.1 × l to 0.2 × l which upon ν C = 0.3 gives the stiffness of the cell body KCel l (Squ ) = 0.14EC l to KCel l (Squ ) = 0.28EC l so:

KCel l (Squ ) 0.21EC l EC ≈ = 0.87 ≈ 0.35 KMat (Squ ) 0.24EM l EM

(18)

i.e. the square cell at myogenic range experiences the matrix almost 3 times as stiff as its own body while for a rectangular cell with S = 15.62:

KCel l (Squ ) × (S + 1/S )/(1 + 1/S ) KCel l (Rct ) = KMat (Rct ) KSqu × KRct (1/S ) KSqu

0.21EC l × (14.74 ) 2.52EC = = ≈1 0.24EM l × (5.12 ) EM

(19)

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A.H. Haji / Journal of Theoretical Biology 486 (2020) 110086

Fig. 7. Location of point A with respect to the lateral sides to compute the contribution of their distributed forces to the displacement at A in a hexagonal cell

Fig. 8. Protrusion length effect on the square cell’s experienced stiffness relative to that of the hexagonal cell

where KRct (1/S)/KSqu is read out from Fig. 6 for 1/S = 0.064 and δ /l = 0.1. The stiffness-based analysis so implies that by reshaping into a rectangle (along with increasing the Young’s modulus) the cell actually tries to reach a compromise between the experienced stiffness of the matrix and that of its own body. 2.3. Hexagonal cell By a square-to-rectangle change we investigated the high aspect ratio effect on stiffness. Here we examine the effect of low aspect ratio by a square-to-hexagon transformation i.e. by increasing the number of the polygon sides and hence, getting closer to the roundness. For a hexagonal cell (Fig. 7) the displacements are:

uH 1 = −2 f 1

√

3l ± δ /2; 0, l/2

uH 2 = 4 f 1 (0, δ /2; 0, l/2 ) √  1 5 3 H u31 = 2 cos(π /3 ) f1 l ± δ /2; l, l 4 4 4 √  1 5 3 + 2 sin(π /3 ) f2 l ∓ δ /2; l, l 4 4 4  √  1 5 3 3 uH = −2 cos ( π / 3 ) f l ± δ / 2 ; l, l 1 32 4 4 4  √  1 5 3 3 + 2 sin(π /3 ) f2 l ∓ δ /2; l, l 4 4 4

Fig. 9. Configuration of a hexagonal cell with dendrites and concentrated tractions at dendrite tips

 √ Under constant-area assumption lHex = (1/3 ) 2 3lSqu and zHex = zSqu so: (20)

where uH is the contribution of the traction band on the side 1 1 to the x-displacement at (0, 0) (point A), uH is the contribution 2 of the side 4, uH 31 is the contribution of the sides 3 and 5 and uH is the contribution of the sides 2 and 6. As before the x32 H H H deflection is H x = |−2 (u1 + u2 + u3 )| which gives the x-stiffness H while K ˜ as KHex_x = F /H = p l δ /  Hex = KHex_y = KHex_x as due to x x the symmetry. For the same parameter values as before we get KHex = 0.2El which shows almost 20% reduction to KSqu . Under constant-area as √ sumption one gets lHex = (1/3 ) 2 3lSqu and KHex = 0.57KSqu . The hexagonal cell, therefore, experiences the substrate significantly softer than the square cell. Besides, Fig. 8 shows that the protrusion length effect is milder for a hexagon compared to a square cell. 2.3.1. Body-to-substrate stiffness ratio for a hexagonal cell Following the same analysis as for equation set (10) the approximate body stiffness for a hexagonal cell can be found as:

√ √ (1 − νC )σC 3l l = ε × 3 l = EC P σC (l × z ) EC z KCel l (Hex ) = = = √ l l 3(1 − νC )

(21)

KCel l (Hex ) =

√ 3 K 3 Cel l (Squ )

(22)

In neurogenic differentiation range i.e. for EM ≈ 1kPa, the cell Young’s modulus is EC ≈ 1kPa (Solon et al., 2007). So the body-tosubstrate stiffness ratio for a square cell equals to:

KCel l (Squ ) 0.21EC l EC ≈ = 0.87 = 0.87 KMat (Squ ) 0.24EM l EM

(23)

assuming the cell thickness 0.15 × l and ν C = 0.3. Interestingly the same ratio is obtained for a hexagonal cell:

√ √ ( 3/3 )KCel l (Squ) ( 3/3 ) × 0.21EC l KCel l (Hex ) EC = = = 0.9 = 0.9 KMat (Hex ) 0.57KSqu 0.57 × 0.24EM l EM (24) However we will see in what follows that by more deliberating the shape and in essence, by growing the dendrites the cell tries to further improve this ratio towards 1. 2.3.2. Hexagonal cell with dendrites For modeling a hexagonal cell with dendrites we first consider the stiffness experienced by the dendrites alone and hence, assume that the tractions are all concentrated at circles of diameter δ at dendrite tips (Fig. 9). The uniform traction distribution over a circle of diameter δ produces a displacement at the center equal to (see

A.H. Haji / Journal of Theoretical Biology 486 (2020) 110086

7

Fig. 11. Dendrite effect in modification of the body-to-substrate stiffness ratio for a hexagonal cell

Fig. 10. Distribution of the contractile nodes (focal adhesions) for a hexagonal cell with dendrites

Appendix B):

uD11 =

 2π  δ /2 0

0

p˜ 2P ν

(1 − ν sin2 (θ ) )drdθ = 1− (25) 2π μ π δμ 2

P = 0.25π δ 2 p˜ being the total traction force. Accounting the contributions from all dendrites leads the total displacement of the dendrite tip 1 to be:

+ 2{uD (x13 , y13 ) cos(2π /3 )

(29)

Now the average stiffness of the cell body reads:

 √ a

3 al + (1 − a ) 3 2d



EC z

(1 − νC ) (30)

√ uH 1 = 2 f 1 ( 3l ± δ /2; 0, (1 − a )l/2 )

(27)

˜ (1 − a )l δ /|−2(uH which gives KHex = F /H + uH )|. The average x = p 1 2 experienced stiffness by the hexagonal cell then reads:

KMat (DendHex ) = aKHex + (1 − a )KDendHex

KDend

(26)

where the traction coordinates are: x12 = (l + d ) sin(π /6 ), y12 = −(l + d ) cos(π /6 ), x13 = 1.5(l + d ), y13 = −(l + d ) sin(π /3 ), x14 = 2(l + d ), y14 = 0. For parameter values (ν = 0.3, δ /l = 0.1, d/l = 1) the stiffness experienced at dendrite tips for a hexagonal cell reduces to one third that of the hexagon without dendrites; KDndHex  √= 0.07El while under constant-area assumption lHex = (1/3 ) 2 3lSqu and KDndHex = 0.18KSqu . The sensitivity of this stiffness to d/l is negligible (not shown) which is an expected result since the deflection caused by a concentrated traction falls off very rapidly by moving away from the vicinity of the force and hence, the major contribution to the deflection at each tip is due to the concentrated traction at the same tip itself. We now refer to the stiffness of the original hexagonal body which should be added to KDndHex . However for the present case where the dendrites are grown at hexagon vertices, the traction nodes are distributed all over the cell membrane Fig. 10). Assume the average length of the dendrite root equals to nδ = al and hence the average length of the traction nodes laid over the hexagon sides equal to l − nδ = (1 − a )l. The experienced stiffness of the substrate by these nodes can be found in a similar manner as before and we use equation set ((20) except that the displacements due to the lateral sides are ignored since the tractions are more concentrated and hence, less transversely-effective for the present case:

uH 2 = 4 f 1 (0, δ /2; 0, (1 − a )l/2 )

2(1 − νC )σC d EC P σC (nδ × z ) al EC z = = = l l 2d (1 − νC )

l = ε × ( 2 d ) =

KCel l (DendHex ) = aKCell + (1 − a )KDend =

uD1 = uD11 + 2{uD (x12 , y12 ) cos(π /3 ) + vD (x12 , y12 ) sin(π /3 )} + vD (x13 , y13 ) sin(2π /3 )} − uD (x14 , y14 )

2.3.3. Body-to-substrate stiffness ratio for a hexagonal cell with dendrites In order to obtain the average stiffness of the cell body the stiffness of the dendrites should be found too. Compared to the cell hexagonal body the dendrites are much softer and hence, a pair of dendrites on the same diagonal of the hexagon is taken as two springs in series for which the stiffness is found as:

(28)

Fig. 11 shows that the body-to-substrate stiffness ratio KCell(DendHex) /KMat(DendHex) equals to 1 for a ≈ 0.12. 2.4. Considerations for osteogenic differentiation We have investigated major shape changes in differentiation including breaking of symmetry in one direction, developing polygonal shape, elongation of cusp corners and formation of dendrites. By reshaping into a rectangle (symmetry break in one direction) the substrate stiffness experienced by the cell increases however the cell body stiffness grows up faster and hence the body-tosubstrate stiffness ratio increases. Such symmetry breaking is a fundamental mechanism selected by the cell which dramatically improves the body-to-substrate stiffness ratio when the cell elasticity (Young’s modulus) is significantly lower than that of the substrate. On the other hand, growing the dendrites is the optimum mechanism selected by the cell to elaborately refine the body-tosubstrate stiffness ratio when the Young’s modulus of the cell is close to that of the substrate. While via these shape changes the stiffness approach has provided acceptable explanations for myogenic and neurogenic differentiations, nevertheless, we may ask if there is any extension which can explain the osteogenic differentiations too. As the osteogenic differentiation takes place on hard substrates, we may expect that the rectangular shape change should take place for osteogenic differentiation in the same manner as for myogenic differentiation. This in fact can be verified observing the shape changes at initial stages of an osteogenic differentiation. The cell on a hard substrate initially breaks symmetry and elongates in one direction while it shifts to another direction later and elongates laterally (Engler et al., 2006).

8

A.H. Haji / Journal of Theoretical Biology 486 (2020) 110086

Fig. 12. Shape (and stiffness) adaptation at neurogenic (EM ~ 1kPa), myogenic (EM ~ 10kPa), and osteogenic (EM ~ 10kPa) differentiation zones

Actually in the osteogenic range the rectangular transformation does not enhance the body-to-substrate stiffness fast enough. This is particularly due to the fact that by increasing the substrate Young’s modulus beyond myogenic range the cell cannot increase its elasticity anymore (EC saturates at 4 to 5kPa) (Solon et al., 2007). This leads to a significantly small EC /EM at osteogenic range (EM > 20kPa) which indeed cannot be compensated even by aspect ratios higher than 16: 1 corresponding to myogenic differentiation. (For a very slender rectangle with 1/S = 0.33 (not shown in Fig. 6), for instance, Eq. (19) gives KCel l (Rct ) /KMat (Rct ) = 0.67 for EC = 4kPa, EM = 20kPa, S = 30, δ /l = 0.05). As the body-to-substrate stiffness ratio is not improved fast enough the cell tries other directions. So the total effect is that the cell develops in 2 dimensions its original polygonal shape. The shape changes to adapt the surroundings stiffness corresponding to neurogenic, myogenic and osteogenic differentiations are summarized in Fig. 12. 4. Conclusion The elasticity-directed differentiation is based on the fact that the differentiation of the stem cells depends on the Young’s modulus of their linear substrate. This paper argues that the elasticity indicator needs to be a more representative one (the stiffness) for the complicated cell-surroundings interaction. In fact, the cell instead of only measuring the elasticity at a local scale (applying the stress-strain relationship) deals with that at global scale (applying the force-stiffness relationship at entire cell-body scale) and particularly includes the geometrical information too. The stiffnessdirected differentiation, hence, actually entails more compatibility to the fact that the cell behaves as a unified whole system and

that its various behaviors such as exertion of the force, measurement of the deflection and transformation into other shapes (and differentiation) are synergistically related to each other. This paper examines the connection between the cells’ experienced stiffness and their geometry and discusses how dramatically it could change by changing the cell morphology. Principle cellular shapes including square, rectangle, and hexagon (with and without dendrites) are taken to argue that the larger the aspect ratio i.e. the further the shape to the original round shape the larger the substrate stiffness experienced by the cell. The substrate stiffness experienced by a slender rectangular cell for some typical aspect ratios due to muscle cells is above six times greater than that of the square cell of the same area. On the other hand, for the hexagonal shape the experienced stiffness reduces to around onehalf of the square cell of the same area, while for hexagon with dendrites (a rough estimation of a neuron’s common shape) the experienced stiffness reduces to one fifth. This explains the differentiation trend from softer to stiffer for the culture conditions where the cell shape is confined by some predefined patterns with small to large aspect ratios. Moreover, for the free culture condition where the cells are allowed to change their shape this study shows that there exists a subtle compromise between the experienced stiffness of the substrate and that of the cell’s own body. At myogenic differentiation range i.e. at substrate Young’s modulus around 10kPa a square cell feels the substrate almost 3 times as stiff as its own body while a rectangular cell with aspect ratio around 16: 1 senses the substrate and its own body alike (body-to-substrate stiffness ratio equals to 1) which suggests why the slender rectangular shape is preferred. As another example it is shown that at neurogenic range i.e. at substrate Young’s modulus around 1kPa the body-to-substrate stiffness ratio is almost the same for the square and hexagon (around 0.9) while by growing the dendrites the ratio is further modified towards 1. It is also worth to say that the substrates stiffnesses are found for a 2D culture condition while inside a 3D matrix the experienced stiffness is larger (see Appendix B). Therefore, for the bodyto-substrate stiffness ratio to be remained in the same range as 2D one expects that the cell body stiffness also should increase in 3D. This can be achieved by increasing the cell thickness which was forbidden by the constant attachment area in 2D case. In fact the 3D extension removes any constraint on the attachment area and hence, the only constraint is on cell volume which does not deny any reshaping with thickness change. Appendix A For the first displacement integral:

 u1 = −

l+δ /2 l−δ /2



l/2 0

− p˜



ν x2



(1 − ν ) + 2 dydx  x + y2 π μ x2 + y2

take y/x = tan(θ ) so:



l+δ /2



tan−1 (l/2x )

p˜ (1 + tan2 (θ ))

 π μ 1 + tan2 (θ )   ν × (1 − ν ) + dθ dx 1 + tan2 (θ )

u1 = −

l−δ /2

0

For 0 ≤ θ < π /2:



(A.1)

tan−1 (l/2x)  (1 − ν ) p˜ u1 = − − ln |sec(θ ) + tan(θ )| π μ l−δ /2 0 tan−1 (l/2x)   ν p˜ + sin(θ ) dx πμ 0 

l+δ /2



A.H. Haji / Journal of Theoretical Biology 486 (2020) 110086

 =−

l+δ /2



l−δ /2

    2 (1 − ν ) p˜  l l  ln  1 + ( ) +  πμ 2x 2x   

ν p˜ l dx = I1 + I2  π μ l 2 + (2x )2

+

=−

(A.2)

Now take w = l/(2x ) so:

    2   l+δ /2 (1 − ν ) p˜  l l  I1 = − ( ln  1 + + dx πμ 2x 2x  l−δ /2   w2   (1 − ν ) p˜l  2 + wdw = ln 1 + w   2w2 π μ w1  w2 (1 − ν ) p˜l −1   = ln  1 + w2 + w πμ 2w 



w1

(1 − ν ) p˜l 1 + dw √ 2 w π μ 1 + w2 w1 In the last integral let w = tan(α ):  w2  (1 − ν ) p˜l 1 1 (1 − ν ) p˜l α2 dw = c sec(α )dα √ 2π μ w 1 + w2 2π μ w1 α1 (1 − ν ) p˜l α = ln |c sec(α ) − cot(α )||α21 2π μ √ w2 (1 − ν ) p˜l  1 + w2 − 1  = ln   2π μ w w1

So:

I1 = −

(A.3)

w1

l+δ /2

ν p˜



l

w2

where w1 = l/(2l + δ ) and w2 = l/(2l − δ ). Now for the third displacement integral:

u3 = 2

 (l−δ )/2  (l+δ )/2

2 p˜ν yx

l 0

4π μ



(x2 + y2 )3

dydx

with the same variable changes:

u3 =

=

 (l−δ )/2  (l+δ )/2

(l+δ )/2

0

l 2 − 1 )dx x

(A.6)

l/2



0

l/2



2 p˜



4π μ x2 + y2

0

(1 − ν ) +



ν x2 x2 + y2

dydx

(B.1)

which is in fact the same as Eq. (5) (with δ = l). The stiffness measured by such a cubic bead then is given by:

p˜l 2 u

(B.2)

(A.5)

KCube = 0.8El

(B.3)

However for cylindrical or spherical beads we expect lower experienced stiffness. The attachment area for a cylindrical bead with diameter l (a circle of diameter l), for example, is smaller than that of a cubic bead of side length l (a square of side length l). So the force is more concentrated for the cylindrical bead and the experienced stiffness is lower. Fig. B1 shows the force distribution over a circle of diameter l. Using Eq. (1) the displacement at the center of the circle (point (0, 0)) due to the distributed force p˜ can be found as:

 u= C



2 p˜



4π μ x2 + y2

(1 − ν ) +



ν x2 x2 + y2

dxdy

(B.4)

tan−1 (l/x )

p˜ν tan(θ )(1 + tan2 (θ ))

where C denotes the circle area. With x = R cos(θ ) and y = R sin(θ ) one gets the corresponding relation in polar coordinate which can be simply integrated as:

tan−1 (l/x )

p˜ν

u=

0

 (l−δ )/2 

1+

For ν = 0.3 one gets:

(A.4)

Using Eqs. (A.2), (A.3) and (A.4):

w1

u=4

KCube =

w1

 w2 (1 − ν ) p˜l   u1 = − ln  1 + w2 + w 2 π μw w √ w2 1 2  1 + w − 1  p˜l  + ln   2π μ  w

1

(

The force-displacement curves are obtained for some typical beads inserted into the matrix and pulled away from a cell (Han et al., 2018). The stiffness of the matrix at various distances to the cell is thus measured. The nonlinear differential stiffness Knl is defined as the slope of the force-displacement curve which increases with applied force F to some saturated force-stiffening limit (Han et al., 2018). However, the cell-matrix interaction does not reach such saturation for low-level force regime which takes place at least during the early stages of the generation/exertion of the contractile force by the cell. Particularly for a typical stem cell, this should be the relevant case for most of its interaction period with the matrix up to complete differentiation. One can, therefore, take Knl at small force limit to get the so-called linear stiffness Klin . For collagen matrix, with differential modulus around 10Pa, Knl is measured 10pN/μm to 20pN/μm over small force domain (Han et al., 2018). To compare this with the analytical model prediction in Section 2.1 a few points should be noted. The model for a distributed force resembling that of a bead is a bit different to a contractile cell. The stiffness measured by the bead can be thought of as due to two semi-infinite surfaces as described in Appendix E. For a cubic bead with a square attachment area to the surface the displacement (at the square center) can be found as:



ν p˜ l w dw √ 2π μ w2 1 + w2

dx = −  2 l−δ /2 π μ w1 l 2 + ( 2x )     α2 ν p˜ l 2 I2 = − tan (α ) + 1 dα α1 2π μ tan (α ) ν p˜l = ln |c sec(α ) − cot(α )| 2π μ √ w ν p˜l  1 + w2 − 1  2 = ln  2π μ  w I2 = −

πμ

(l−δ )/2 p˜ν  2  =− ( l + x2 − x ) πμ (l+δ )/2 ⎛  ⎞  2   2 p˜ν ⎝δ − l 2 + l + δ + l 2 + l − δ ⎠ =− πμ 2 2

Now for I2 with the same variable changes we have:



(l+δ )/2

Appendix B

w2

 w2 (1 − ν ) p˜l   ln  1 + w2 + w 2 π μw w √ w12 (1 − ν ) p˜l  1 + w2 − 1  + ln   2π μ w

 (l−δ )/2 p˜ν

9

dθ dx  3 π μ (1 + tan2 (θ ))

πμ

sin(θ )dθ dx

 θ 0

0

R

p˜ 2π μ

(1 − ν sin2 (θ ))drdθ =

p˜R

μ



1−

ν

2

(B.5)

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A.H. Haji / Journal of Theoretical Biology 486 (2020) 110086

Appendix C The stiffness experienced by the cells in assembly In our survey for the geometric effects in cell-substrate mechanics, here we zoom out from the cellular scale (cell shape) to the extracellular scale and investigate the cells configuration effect. The appropriate geometric parameter which characterizes the cells configuration would be exploited from the concentration of the cells; the average distance s between the cells which would have a pivotal role in determining the cells experienced stiffness. For a uniform distribution of the cells (see Appendix D):

1 s= √ n

Fig. B1. Square (a) and circular (b) distribution of a tangential force on a substrate.

Substituting for p˜ = P/(π R2 ) where P is the total traction force gives:



P ν u= 1− π Rμ 2

(B.6)

and the stiffness reads:

KCyl =

P π = El u 4(1 + ν )(1 − ν /2 )

(B.7)

upon substituting R = l/2 and μ = E/(2(1 + ν )). With ν = 0.3 we have KCyl = 0.71El which is 11% lower than KCube given by Eq. (B.3). For the spherical beads (as those used in Han et al. (2018)) we expect the stiffness to be measured even lower than KCyl . The reason is that the effective attachment area of a spherical bead to its surrounding matrix is less than that of a cylindrical bead of the same size and hence, the force distribution turns to be more concentrated over a spherical bead. Add to this the point that the collagen fibers cannot uniformly attach to the whole area of the bead. We may assume 20% reduction as due to each of these effects. The measured stiffness of a collagen matrix by a spherical bead is therefore approximated as:

K = 0.8KSphr = 0.8 × 0.8KCyl = 0.45El

(C.1)

where n is the cells concentration in 2D (number of the cells per unit area). In order to find the substrate stiffness experienced by a cell in assembly the stiffness of the tensioned (intracellular) portion of the substrate between the neighboring cells should be added to the stiffness of the under cellular portion KU (see Appendix E). The procedure to calculate the intracellular stiffness is just as that for the under cellular portion except that the cell traction in the y-direction is removed. For the x-displacement at A(0, 0) (taken at right side of the cell as shown in Fig. C1) due to the concentrated traction at (s − l, 0 ): u1 = 2 =2

p˜ I f1 (s − l ± δ /2; 0, l/2 ) p˜  s−l+δ /2  l/2 2 p˜ I s−l−δ /2

4π μ x2 + y2

 (1 − ν ) +

ν x2 x2 + y2

 dydx

(C.2)

with p˜ I being that part of the cell traction which applies to the intracellular substrate portion. However, u1 in this case depends √ on two geometric parameters: δ /l and q = s/l = 1/(l n ), the latter would be called the network configuration parameter. The xdisplacement at (0, 0) due to the concentrated traction at (0, 0) is given by:

u2 = 4 f1 (0, δ /2; 0, l/2 )

(C.3)

as before and the total displacement can be found as uA = u1 − u2 while transverse effects due to the neighboring cells are ignored. The total deflection of the intracellular segment is 2uA and the corresponding stiffness reads:

K=

p˜ I l δ |2uA |

(C.4)

Following a similar discussion as for KU , the intracellular stiffness KI relates to the stiffness K and the semi-infinite surface stiffness KS :

(B.8)

Substituting for the bead diameter and matrix modulus l = 4.5μm, E = 10Pa we get K = 20.4 pN/μm which is quite comparable to the experiment (Han et al., 2018). A final notion is that the 3-dimensional extension of the present analytical model can be achieved simply by assuming two semi-infinite spaces attached together i.e. a thick 3D matrix with all around surfaces fixed. The experienced stiffness inside such a matrix and far enough from boundaries (top, bottom and peripheral surfaces) should be twice as much as the stiffness experienced on the semi-infinite case. However, the most common 3D case takes place inside a semi-infinite matrix where the experienced stiffness depends on the distance (depth) to the top surface. At small depth the stiffness is obviously close to the experienced stiffness on the surface (20.4pN/μm). By increasing distance to the surface the stiffness increases, however, significantly less than that experienced inside a completely fixed matrix (40.8pN/μm).

0



KI = K −

KS 2

(C.5)

Fig. C1. Tensioned intracellular portion and two compressed semi-infinite portions (left and right of the intracellular portion)

A.H. Haji / Journal of Theoretical Biology 486 (2020) 110086

11

Fig. C4. Density effect on normalized average stiffness experienced by the cells (ν = 0.3(

Fig. C2. Cellular network on substrate with parallel springs connecting the middle points of the under cellular and intracellular portions

20 0 0 cell/cm2 ( ~ 3 × 104 to 9 × 104 cell/cm3 ) up to relatively high densities around 250 0 0 cell/cm2 ( ~ 4 × 106 cell/cm3 ) (Saha et al., 2008; McBeath et al., 2004; Xue et al., 2013; Murphy et al., 2012; Rowlands et al., 2008; Sharma and Snedeker, 2012). For some typical stem cell size equal to 10μm the high density region locates at s/l = 6.3 where the linear-elasticity analysis shows no significant increase in the experienced stiffness (Figs. C3 and C4). However it should be noticed that the elastic model predictions are just valid for low density condition when the interactions of the cells are small. Particularly at the vicinity of the cells the linear elasticity predictions might be subject to significant errors due to the nonlinear effects (stress/strain hardening). Such effects have significant effect on the total experienced stiffness by the cells (Han et al., 2018). Appendix D

Fig. C3. Density effect on normalized average stiffness experienced by the cells (δ /l = 0.1(

For a uniform distribution of the cells with corresponding concentration (at surface) equal to n, the number of the cells inside a square of size x reads:

As far as the intracellular distance is significantly larger than the cellular dimension, i.e. for small enough densities we can neglect this intracellular stiffness. For high level density on the other hand, the under cellular and intracellular portions should be considered at the same time. Assume the location of the cells do not change significantly and particularly that there is minor relocation due to cell locomotion and diffusion so that the deflections of the equivalent under- and intracellular springs are almost the same in magnitude but opposite in direction (Fig. C2). The half-springs connecting the middle points of the succeeding under cellular and intracellular portions, form two springs in parallel for which the equivalent stiffness reads Kl/2 = 2(KI + KU ) and so the cells experience the matrix as stiff as KN = Kl = 0.5Kl/2 = KI + KU . which is the average experienced stiffness for the cells in assembly. Dividing by the singularcell experienced stiffness, the normalized form reads:

N = n(x )2 .

Ke =

KN KSqu

(C.6)

Figs. C3 and C4 show the density, Poisson’s ratio and protrusion length effects on this stiffness. At sparse conditions (large s) regardless of the changes in δ /l and ν the stiffness KN as expected tends to the sparse-condition stiffness KSqu corresponding to a singular cell on an infinite domain. On the other hand, at large densities (small s) the stiffness due to the intracellular portion takes effect and contributes to the total stiffness. However the stiffness increases just twice (by decreasing s/l) only at relatively very large density domain. In fact the experiments are generally conducted at very low (sparse condition) densities usually around 10 0 0 to

(D.1)

If there are Nx cells in each column and row, by subtracting the number of common cells (at the edges and corners) from Nx2 we may write:

N = (Nx − 1 )2 ,

(D.2)

which upon (D.1) gives:

√ Nx = x n + 1.

(D.3)

Therefore for the average mutual distance s we have:

s=

x 1 = √ . (Nx − 1 ) n

(D.4)

For one-dimensional cellular dynamics in the x-direction the cells traction force is also in this direction. At any cross section y perpendicular to the x-direction (at the right side of a square cell with dimensions x = y) there are Nx cells exerting traction force on the substrate. So the density of the cells pulling arms (protrusions) at each cross section also can be written as:

nx = (Nx − 1 )/(x ) =

√

n.

(D.5)

Appendix E Contribution to the substrate stiffness experienced by the cell comes from two origins: 1) the under cellular portion of the substrate and 2) the semi-infinite spaces left and right-hand side of the cell which act like two springs in parallel to the under cellular spring as represented in Fig. E1. Let KU the stiffness of the

12

A.H. Haji / Journal of Theoretical Biology 486 (2020) 110086

As there are two semi-infinite springs in parallel at this location (Fig. E2), this stiffness equals to 2KS :

KS =

p˜l δ 2u2

(E.4)

Substituting from Eqs. (9) and (E.3) in (E.1) now gives KU . For typical parameter values ν = 0.3 and δ /l = 1/10, we get KS = 0.19El and KU = 0.15El. References

Fig. E1. Compressed under cellular portion and two tensioned semi-infinite portions (left and right side of the cell).

Fig. E2. Equivalent spring model for two semi-infinite surfaces subject to a local distributed force.

full-length under cellular portion, so the stiffness at half-length is 2KU . This should be added to the half-plane or semi-infinite surface stiffness KS to give the total stiffness. For static (no relocation) case which is quite reasonable at sparse condition, the centerline remains fixed. The deflection of the half-length under cellular portion thus equals to x /2 and the total stiffness equals to 2KSqu . The spring model for a singular cell thus leads to the following relation:

2KSqu = 2KU + KS

(E.1)

and to obtain KU it just remains to calculate the semi-infinite surface stiffness KS . As described above there can be considered two half-planes to the left and right side of a tangential force F = p˜l δ distributed over a finite rectangle l × δ on the half-planes’ edge. Note that for such structural parts (two half-planes) the equilibrium condition is guaranteed by the far field zero-displacement in the Cerruti’s solution. The deflection u at the center of the rectangle is given by the following equation which is the same as Eq. (5):

 u=4

+δ /2 0

 0

l/2

2 p˜



4π μ x2 + y2

 (1 − ν ) +

ν x2 x2 + y2

 dydx

(E.2)

and hence, the experienced stiffness at (0, 0) reads:

K=

p˜l δ u

(E.3)

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