Biomaterials 35 (2014) 8078e8091
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Biomaterials journal homepage: www.elsevier.com/locate/biomaterials
The mechanisms of fibroblast-mediated compaction of collagen gels and the mechanical niche around individual fibroblasts Zhonggang Feng a, *, Yusuke Wagatsuma a, Masato Kikuchi a, Tadashi Kosawada a, Takao Nakamura b, Daisuke Sato b, Nobuyuki Shirasawa c, Tatsuo Kitajima d, Mitsuo Umezu e a
Graduate School of Science and Engineering, Yamagata University, Japan Graduate School of Medical Science, Yamagata University, Japan Department of Anatomy and Structural Science, Yamagata University, Japan d Electronic Systems Engineering, Malaysia-Japan International Institute of Technology, Malaysia e Integrative Bioscience and Biomedical Engineering, Graduate School of Waseda University, Japan b c
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 March 2014 Accepted 24 May 2014 Available online 26 June 2014
Fibroblast-mediated compaction of collagen gels attracts extensive attention in studies of wound healing, cellular fate processes, and regenerative medicine. However, the underlying mechanism and the cellular mechanical niche still remain obscure. This study examines the mechanical behaviour of collagen fibrils during the process of compaction from an alternative perspective on the primary mechanical interaction, providing a new viewpoint on the behaviour of populated fibroblasts. We classify the collagen fibrils into three types e bent, stretched, and adherent e and deduce the respective equations governing the mechanical behaviour of each type; in particular, from a putative principle based on the stationary state of the instantaneous Hamiltonian of the mechanotransduction system, we originally quantify the stretching force exerted on each stretched fibrils. Via careful verification of a structural elementary model based on this classification, we demonstrate a clear physical picture of the compaction process, quantitatively elucidate the panorama of the micro mechanical niche and reveal an intrinsic biphasic relationship between cellular traction force and matrix elasticity. Our results also infer the underlying mechanism of tensional homoeostasis and stress shielding of fibroblasts. With this study, and sequel investigations on the putative principle proposed herein, we anticipate a refocus of the research on cellular mechanobiology, in vitro and in vivo. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Collagen Fibroblast Hydrogel Modelling Cell culture
1. Introduction Fibroblast-mediated compaction of collagen gels attracts widespread attention in studies related to wound healing [1,2], cellular fate processes [3e6], and regenerative medicine [7e11]. Remarkably, however, more than three decades after this phenomenon was first described in detail by E Bell et al. [12], the underlying mechanism and the micro mechanical environment around individual cells still remain obscure. Fibroblasts in newly formed collagen gels are characterised by their striking ability to protrude pseudopodial extensions e as long
* Corresponding author. Graduate School of Science and Engineering, Yamagata University, Johnanmachi 4-3-16, Yonezawa City, Yamagata Prefecture 992-8510, Japan. E-mail address: [email protected] (Z. Feng). http://dx.doi.org/10.1016/j.biomaterials.2014.05.072 0142-9612/© 2014 Elsevier Ltd. All rights reserved.
as several tens of mm within a few hours. These pseudopodia possess a microtubule core and actin-rich tips [13]; their extension has been demonstrated to be integrin mediated [14], and Rhofamily kinases are involved in the regulation of the actin-myosin dynamics of the cytoskeleton [5,15]; and, they ultimately constitute a network via intercellular contacts [16,17]. In this process, collagen fibrils adjacent to the pseudopodia (or in the front of the tips thereof) become aligned with the long axis of the pseudopodia, which induces a local compaction of the network [18,19]. Morphological investigations have generally suggested that this compaction is caused by cells exerting a traction force on the attaching collagen fibrils, which in turn results in the accumulation of fibrils around the cells. However, recent studies have challenged this opinion [20]. At present, the anisotropic biphasic theory (ABT) [21] is the only sophisticated theory on this subject. This approach regards the
Z. Feng et al. / Biomaterials 35 (2014) 8078e8091
collagen gel as a continuum biphasic material wherein the cells are simply a component of the network phase, which is phenomenologically modelled as a Maxwell-like fluid. The factor that drives gel compaction in the governing equations of ABT is a multiplicative term comprising cell-generated stress, cell concentration, and cell orientation. In other words, ABT explains collagen gel compaction as an integrated effect involving cell traction, cell migration and proliferation, and cellular and fibrillar reorientation. The deeply interactive nature of the various biological and physical elements of the gel compaction process deters the application of a simple cause-and-effect description, and it would be appropriate to approach the problem with due presumption of complexity [22e26]. The micro mechanical environment around individual cells undoubtedly plays a vital role in gel compaction, initially drawing attention due to its close relation to the physiological and pathological behaviours of wound healing [27,28]. And recently, the importance of this issue was raised in studies of the stem cell's niche [29e32]. The force exerted by individual fibroblasts in collagen gels has been measured in the range of 0.1e450 nN by means of direct measurement in gel slabs [33e36] or extrapolation through the deformation of the substrate [37e40]. Although these data show that the cellular contractile force is variable and sensitive to the extracellular environment and culture conditions, the overall micro mechanical environment has not been fully elucidated, quantitatively. Therefore, despite the multitude of studies based on this model tissue, there is still a lack of clear answers to the following important questions. What is the physical process by which cells exert traction force on the surrounding fibrils while compacting the entire construct? While the effects of cellular random migration and fibrillar orientation are spatially limited to the immediate vicinity of the populated cells, the majority of the macro-compaction of the gels is attributable to those fibrils distal to the cells. This begs the question: In what state are those distal fibrils and what roles do they play in the process? A specific value of traction force results from the interplay between cells and the surrounding collagen fibrils. The mystery herein is the underlying regulatory mechanism by which a cell acquires so particular a value from within the broad range of its force-generating capacity. These questions motivate us to try to disentangle the convolution of cause-and-effect in the hope of uncovering the initiating agent of collagen gel compaction, and to expose the micro mechanical niche around individual fibroblasts, by means of mechanical modelling. We particularly focus on the analysis of the collagen network because this, as the passive partner to the active populated cells in the compaction, is much more stable and easily identified. Although the fibril network itself is also a complex structure composed of random cross links and entangled fibrils, it is more readily elucidated than the populated living cells whose behaviour it exactly reflects, as the cell-fibrillar network provide the only primary mechanical action-counteraction coupling within the system [41]. Our approach is based on the statistics of fibrillar networks [42] and is inspired by network alteration theory [43]. We introduce the classification of fibrils into three different types according to their roles during the compaction process and deduce the respective governing equations that exactly describe the mechanical behaviour of each type. In particular, a putative principle based on the stationary state of the instantaneous Hamiltonian of the mechanotransduction system is employed in the deduction. We justify our theory by applying it to two representative fibroblastcollagen gel compactions: free-floating and quasi-uniaxial constraint. In the course of this justification, intriguing findings about the micro mechanical environment around individual fibroblasts in these two systems are obtained. Finally, we discuss the
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implications of our findings in this model tissue with regard to its predicated equivalence to the relevant cellular biology and connective tissue physiology. 2. Experimental materials and methods 2.1. Gel formation and compaction measurement Two types of compactions, free floating (FF) and quasi-uniaxial constraint (UC) (Fig. 1), were experimentally investigated in this study. Modelling was evidenced and verified by the experimental data from these gels, and in turn revealing the (micro) mechanical niche around individual fibroblasts within the gels. Fabrication of the two types of fibroblast-collagen gels was as described in our previous studies [9,10]. Briefly, type I collagen (extracted from rat tail) and dermal fibroblasts (explanted from Wistar rat embryos and subcultured to 5 passages) were mixed with DMEM solutions and poured into two types of casting moulds (Fig. 1A,B). Each gel had a 1.0 ml initial volume; cell number and collagen mass are listed in Table 1. For UC gels, two T-shaped stainless steel wires were initially placed, one anchored to the well edge and the other connected to a vision-based micro force sensor (described below in Measurement of the constraint force in UC gels), for measuring the constraint force during gel compaction. The mixtures were allowed to gelate in a 5%CO2, 37 C incubator for 40 min. After gelation, the FF gels were displaced from the dish bottom by adding culture medium, resulting in a free-floating culture. The ends of the UC gels wrapped spontaneously around the T-shaped anchors. After culture medium was added to float the gels, compaction along the anchor direction was essentially constrained to the region between the two anchors, while compaction could occur freely in the orthogonal directions. As a result, the gels compacted to a string-like shape. The gels were cultured in 6 ml of DMEM supplemented with 1% penicillinestreptomycin and 10% foetal bovine serum (FBS) for one week. Gel size was obtained by image processing of gel pictures using Adobe Photoshop CS4, and compaction was delineated as the ratio of the gel area (for the FF gels) or of the gel width (for the UC gel) to the corresponding initial values. A surfacemarked cover glass was placed on the gels when gel thickness was measured, and the distance between the lower surface of the cover glass and the bottom of the culture dish was measured by means of an inverted light microscope. 2.2. Measurement of the elasticity of FF gels under external compression Gel was immersed in phosphate buffer saline (PBS) and settled on the stage of an analytical balance. A controlled press moved downward to compress the gel at constant speed. After the compression strain reached 5%, the compression was halted and stress relaxation occurred. The compression/relaxation profile was regressed via a model composed of two parallel Maxwell fluid models (details in Supplementary information S1). The elasticity was regarded as owing mainly to the collagen fibrillar network, and was used to verify the theory of fibrillar bending (see 3. Modelling Theory). 2.3. Measurement of the constraint force in UC gels We developed twenty vision-based micro force sensors and individually calibrated them. The principle of these sensors is that transverse force can be investigated by measuring the deflection of glass fibre cantilever through microscope (details in Supplementary information S2). Each sensor was settled onto one UC gel mould (as shown in Fig. 1B) and the constraint force during the gel compaction can be measured. 2.4. Observation of cellular morphology in gels Following the addition of 0.3 ml acetic acid (99.7%, Sigma) to the culture dishes of 1 mg collagen-0.9 million fibroblasts gels (with a confirmed decrease in pH to 3.20) and incubation in a 37 C incubator for 30 min, the collagen gel was dissolved. This allowed the morphology of the embedded fibroblasts to be clearly observed under a biological inverted microscope. Some of the cells that had formed a pseudopodial network within the gel could be separated by gentle pipetting. Pictures were taken of cells with pseudopodial processes, and measurements were made using Adobe Photoshop CS4; these included the diameter of the cellular soma, and the diameter and length of the pseudopodial processes. From these data, cellular surface area could be estimated. 2.5. Observation of collagen fibril morphology by scanning electron microscopy (SEM) SEM samples for FF gel, UC gel, and cell-free gel were prepared by using standard protocol for biological tissues. 2.6. Observation of the expansion of the gel immediately after cellular necrosis This experiment was to demonstrate that the nature of the fibrillar bending as gel compaction is of elastic. Cell necrosis was induced by adding 0.1 ml of 1 N sodium hydroxyl to the FF gel dishes containing 0.9 million cells in 1.0 mg collagen at 8 h culture. Then, gel diameter was measured by means of an inverted microscope.
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Fig. 1. Two types of fibroblast-collagen gels were investigated in this study: (A) a gel under free-floating compaction, FF gel; (B) a gel under uniaxial constraint compaction, UC gel. Traction force during compaction is measured by a vision-based micro force sensor as described below.
2.7. The coefficient of data determination R2 In this study, the coefficient of data determination R2 was defined as P 2 0 0 R2 ¼ 1 N1 N n¼1 ððxn xn Þ=xn Þ ; where xn are the experimental data, xn are the corresponding theoretical regression values, and N is the number of experimental data.
Nevertheless, there remained numerous randomly orientated fibrils among those stressed e many connecting the parallel, stressed fibrils (Fig. 2H). 3.2. Structural elementary model of the compaction
3. Modelling theory 3.1. Evidences of the modelling The structural elementary model of collagen gel compaction developed in this study is supported by the morphological observations on populated fibroblasts and collagen fibrils. Therefore, some of the experimental results related with these observations should be firstly exhibited herein. Fig. 2 shows the evidences of the model from the cellular and fibrillar morphology. Just after 2 h of culture almost all fibroblasts protruded pseudopodial processes to the surrounding collagen fibrils (Fig. 2A). At 4 h, the processes were much longer and cells had gathered closely together (Fig. 2B). At 8 h, the processes contacted each other, forming a 3-dimensional framework (Fig. 2C). From 8 h onwards, most of the cells had merged multiple pseudopodia, assuming a dendritic polar shape (Fig. 2D). In the fibrillar aspect, fibrils in the cell-free gel were relatively straight (Fig. 2E), whereas those in the fibroblast-compacted gel were curved (Fig. 2F) e evidence that almost all fibrils ought to be stretched at the beginning of gel-compaction in case, and finally most fibrils are actually bent as a consequence of gel compaction. The morphology of the fibrillar network in the UC gel was characterised by the linear alignment of the stressed fibrils parallel to the constraint axis (Fig. 2G).
Z
Table 1 Gel compositions used in this study. Collagen:
0.5 mg
ðG þ T þ BÞ dsu 1.0 mg
1.5 mg
Cells ( 106) Free floating (FF) 0.3 0.9 1.5 Uniaxial constraint (UC) 0.3 0.9 1.5 (initial volume ¼ 1.0 ml)
Supported by the above evidences, we abstracted the whole gel into two interacting elementary units (Fig. 3), each of which consists of one central cell and the surrounding collagen fibrillar network. Collagen fibrils are classified into three types: bent (curved lines), stretched (straight lines), and adherent (shadow around the cell). The mechanical state in the horizontal direction is depicted here. T and B are the stretching force and bending force in the corresponding fibrils, respectively. G is the boundary force. In this model, the compaction process begins as the cell protrudes pseudopodia that adhere to adjacent collagen fibrils. Among these adherent fibrils are some with the correct length potential for sustaining the traction force exerted by the two respective pseudopodia, rooted in neighbouring cells. Almost all fibrils have become stretched at the beginning of compaction can be assumed since they are then rather straight (Fig. 2E and verified by the results: Fig. 16B). Due to the existence of a free (i.e., moveable) boundary, the traction force results in a loss of mechanical equilibrium in the direction normal to the boundary; consequently, cells near the boundary are compelled to move closer to those at more interior locations. This movement bends the majority of the fibrillar network. As long as a cell continues protruding or extending its pseudopodia, the loss of mechanical equilibrium would continue until the fibrillar network between the two cells becomes stiff enough to resist the traction force. The equation governing the movement of the elementary unit is then given by Newton's second law,
B B B
B B B
B B B
B
B B B
B
Z
Rc dsc þ Rf dsf
dv ¼ mc þ mf dt
(1)
where: G, T, and B are the boundary force, stretching force, and bending force, respectively; Rc and Rf are the fluid resistance on the cell and the fibrils, respectively; mc and mf are the mass of the cell and the fibrils, respectively; v is the movement velocity, and t is the time. The inertial term on the right side of Eq. (1) is so small as to be negligible. Evaluation of Rc and Rf by using Lamb's formula reveals that they too can be neglected. Consequently, the situation becomes a static problem:
Z. Feng et al. / Biomaterials 35 (2014) 8078e8091
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Fig. 2. Morphological observations of populated fibroblasts by means of acetic acid dissolution (AeD), and of collagen fibrils by means of scanning electron microscopy (EeH). Fibroblasts in FF gel just after 2 h of culture (A), after 4 h of culture (B), after 8 h of culture (C), and cells separated by lightly pipetting from cellular framework after 8 h of culture (D). Fibrillar networks in the cell-free gel (E), in the fibroblast-compacted FF gel (F), and in the UC gel (G, H).
Fig. 3. Structural elementary model of gel compaction.
GþT þB¼0
(2)
3.3. Theory of fibril mechanics in compacted gels In this study, we originally classified the fibrils into three types e bent, stretched, and adherent e and deduced the respective equations governing the mechanical behaviour of each type as below.
3.3.1. Fibril bending force Fig. 4 illustrates the concept of fibril bending during gel compaction. l is the average size of the fibrillar network mesh; ε is the compaction strain; and q is the bending displacement. Note that affine deformation is assumed. Compaction is effected by the bending force B in the medial fragment, which deforms the horizontal fibril, rather than by buckling of the medial fragment itself. Estimation of l is based on the statistical theory of fibre packing developed by S Toll [42], which deals with realistic random fibrillar networks and yields the formula
l¼
Fig. 4. Mechanical schematic for modelling a bent fibril: the compaction displacement, vertical in this schematic, induces bending of the horizontal fibril fragment.
pr 4ff
(3)
where f is the fibril volume fraction, f is a scalar invariant of fibre orientation distribution (equal to p/4 or 2/p for a 3D or 2D random orientation, respectively), and r is the fibril radius. The bending force B is defined as
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B¼
Z. Feng et al. / Biomaterials 35 (2014) 8078e8091
vE vq
(4)
where E is the bending energy. For the semi flexible collagen fibril [44],
E¼
1 k QA 2 B
I
k2 ds
(5)
where kB is Boltzmann's constant (1.380658 1023 J K1), Q is absolute temperature, A is the persistent length of the collagen fibrils, k is the bending curvature, and s is the contour distance from one end. To calculate the curvature k we adopted network alteration theory [43], which regards the kinematics of a filamentous network as a serially formed structure that may be substantially unlike the original one. By this concept, the compaction of the fibrillar network is stepwise. At each step, the network is reformed so as to acquire new values of the network state parameters f and l. The amount of bending in each step is small (as in the gently curved fibrils shown in Fig. 2F), and can be approximated by a sine curve (in accord with Ref. [45]). Thus,
1 k QA 2 B
E¼
¼
I
1 k QA 2 B
00
0
y
(6)
2
1 þ y02
5=2 dx
where y ¼ q sinðp x=2lÞ ¼ ε l sinðp x=2lÞ is the orthogonal coordinate of the fibril, x is the horizontal coordinate, and primes denote the derivative with respect to x. Substituting y into the above equation and recognising it is a small quantity, we can approximate deformation energy as
1 Ez kB Q A 2
Z2l y
00
5 1 y02 2
dx
¼
p4 kB Q A ε2 32 l
p4 kB Q A ε 2 32 l
Hb ¼
1 k QA 2 B
I k2 ds
(11)
He ¼
1 2
I
Gp r 2 ðs=s0 1Þ2 ds
(12)
where G is the elastic modulus of the fibril, s is the contour distance from one end, and s/s0 represents the elastic extension ratio at s. Hq is the work done by the bending force:
Hq ¼ Bq
(13)
where q is the transverse displacement due to the bending force B. Ht is the traction energy exerted by cells that stretch the fibril at its two ends:
(14)
(7)
Therefore, the average bending force at the current step can be expressed as
B¼
(10)
Hb is the bending energy, again calculated by
Ht ¼ T 2l 2l0
0
p4 kB Q A q2 z 3 32l
H ¼ Hb þ He þ Hq þ Ht
He is the energy of elastic elongation, which can be approximated by
k2 ds Z2l
generate traction force arbitrarily but rather in accord with the surrounding environment. As shown in Fig. 5A, the fibrils are stretched by cells from either end, as in a “tug-of-war”, yet are under the influence of a transverse force e the bending force that is applied by the fragments of the random portion of the fibrillar network connecting the stressed fibrils (as evidenced in Fig. 2H). This bending force, as formulated by Eq. (8), is stored in the random fibril network and provides the environment that surrounds the cell while exerting traction force on the stretched fibrils. We analysed a representative fragment in the stretched fibril, as shown in Fig. 5B, in order to discover the relationship between the traction force and the bending force. The effective Hamiltonian for the fragment may be written as the sum of four terms [49]:
where 2l0 signifies the initial end-to-end length of the fragment. To find the relationship between traction force T and bending force B, we consider the fragment to be initially straight (as evidenced by Fig. 2E), before being bent by force B and stretched by force T. The curve of the fibril is approximated as a sinusoid [50] with amplitude q. Thus the contour length after deformation is
(8)
In addition to computation of the bending force applied to the elementary unit, the above equations also allow estimation of the elastic modulus of the gel (denoted by X) under external compression, during its compaction e as long as ε is equated to the external compression strain and bending force to the external pressure:
X¼
p4 kB Q A 1 2 32 l
(9)
3.3.2. Stretching force Generally speaking, the traction force is generated in the cytoskeleton and transmitted to the stretched fibrils via cellular mechanotransduction [46e48]. The proposed concept for deducing the traction force applied to each stretched fibril is that cells do not
Fig. 5. Depiction of a stretched fibril (A): the fibril is stretched by the traction force exerted by two neighbouring cells under the influence of a transverse bending force, which originates from the random fibrillar network. Free body diagram of a representative fragment within the stretched fibril (B).
Z. Feng et al. / Biomaterials 35 (2014) 8078e8091
S¼
#12 Z2l " pq px 2 p2 q2 cos dx ¼ 2l þ 1þ 8l 2l 2l
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(15)
0
If we assume the elongation within the fibril to be uniform, then
1 2
He ¼
2 s Gp5 r 2 q4 1 ds ¼ 3 s0 256l
I Gp r 2
(16)
since s0 ¼ 2l. However, q cannot be set by the affine condition as was done for the random fibrillar network. Here, we propose an important hypothesis that the cells stretch the fibrils in such a way as to achieve stability of the Hamiltonian, that is
dH¼0
(17)
where dH is the variation of H as the fibril is stretched by D at the ends. Energy conservation during bending leads to
Fig. 6. Alteration of cellular surface area during culture in collagen gel. Squares mark the experimental estimates (n ¼ 10). The solid line is derived from Eq. (23).
Bq ¼ Hb þ He
The adhesion force between the cell and individual adherent fibrils may be evaluated as
¼
kB QAp4 q2 3
32l
þ
Gp5 r 2 q4
(18)
3
Thus, 3
256l B ¼ 8p4 kB QAq þ Gp5 r 2 q3
(19)
As the fragment is stretched by the cell, extending the ends by D, the corresponding change in H is
1 q 1 q3 kB QAp4 3 dq þ Gp5 r 2 3 dq þ TD dH ¼ Bdq þ 16 64 l l
(20)
Since by Eq. (15),
dq 4l z 2 D p q
(21)
Substituting Eq. (21) into Eq. (20) and apply Eq. (17), we obtain
T¼
4l k QAp2 Gp3 r 2 q2 Bþ B 2 þ 2 2 p q 4l 16l
(22)
Thus, combining Eqs. (19) and (22) we can obtain the stretching force sustained in individual collagen fibrils. 3.3.3. Fibrils adherent to the cell and evaluation of the fibrillar adhesion force To estimate the total length of the fibrils adherent to the cell, we measured the change in cellular surface area in the cellular morphology images. Fig. 6 shows that the following formula can satisfactorily fit the measurement data:
Sc ¼ 4p r02
a þ b tegt a þ tegt
(23)
where r0 is the initial radius of the cell, t is the culture duration, and a, b, and g are constants. The adhesion of collagen fibrils is regarded as a two dimensional fibrillar web, modelled as a l l mesh wrapped on the cellular surface. Therefore, the total length of the adherent fibrils may be estimated as
Lad ¼
2Sc l
2
t ¼ Bc
256l
(24)
l Sc
(25)
The total bending force Bc can then be interpreted as the resistance to cell movement, distributed between the adherent fibrils. 3.4. Computation procedure In a practical computation, the first step is to calculate fibrillar mass and initial volume in the elementary unit. Then, from the measurement of gel compaction, the unit volume during compaction can be obtained. At each compaction step, the cellular surface area is computed along with the masses of the adherent fibrils, the stretched fibrils, and the bent fibrils. Following this, Eq. (3) is applied to calculate the average length of the bent fragment l. For the FF compaction, by Eq. (2) traction force is equal and opposite to bending force in any direction, because the force at the boundary is zero. For the UC gels, the traction force is equal to the bending force in the lateral direction with the free boundary, whereas it is equal to (GþB) in the constraint direction. The values of the fundamental constants and parameters used in the computations are listed in Table 2. 4. Results 4.1. Fibroblast-mediated collagen gel compaction Experimental results of gel compaction with different gel contents (Table 1) under free floating and constraint conditions are Table 2 Values of the fundamental constants and parameters in the model. Time interval (h)
r0b (mm)
rb (mm)
Tb (K)
Ac (mm)
Gd (GPa)
ab
bb
gb
1.0a
9.5
0.025
310
120
0.4
50
6.0
0.25
a
Gel compaction is several percent over this time interval, coinciding with the result of the gel expansion experiment. Contacts at the fibrillar intersections are required to be stable over a comparable period. b These values were measured as part of this study. c The persistent length of the collagen fibrils was determined by estimating SEM of gels without cells and referred to Refs. [44,51,52]. d The elastic modulus of the collagen fibrils was determined by referring to Refs. [53e55].
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Z. Feng et al. / Biomaterials 35 (2014) 8078e8091
shown in Figs. 7e10. We found that the following formulae can fit the experimental measurement in good agreement.
J2D ¼
a þ bt aþt
J1D ¼
a þ bt aþt
(26) 12 (27)
where J2D and J1D are the area ratio for free-floating compaction and width ratio for constraint compaction, respectively. a and b, listed in Table 3, are constant with respect to each cell number and initial collagen concentration (ICC); t is the culture duration in hour. Summarily, dramatic compaction occurred within the first 12 h (Figs. 7 and 8). Volume compaction ratio was proportional to ICC, but reverse related with cell number as the other condition keeping constant (Figs. 9 and 10). Compaction of UC gels was consistently more pronounced than that of FF gels (Fig. 10). 4.2. Verification of the model Verification of the structural elementary model was provided via the results from the experiments of gel expansion, elastic measurement, and constraint force measurement. In the first, we observed ca. 3% expansion in the diameter of FF gels after inducing necrosis of populated fibroblasts (Fig. 11). This shows that the fibrillar bending is elastic, which is a prerequisite of our theory to compute bending force and is in keeping with network alteration theory; specifically, this allowed the gel compaction to be analysed as a series of small, stepwise deformations. To demonstrate that the observed expansion was not due the change of pH of the medium caused by addition of sodium hydroxyl, we added acetic acid to the gel culture medium; expansion was again observed, and to the same extent. Particularly notable was that cell-free gels subjected to these conditions underwent contraction rather than expansion. In the second experiment, we measured the elastic modulus of FF gels and compared it with the prediction in our theory by Eq. (9). Fig. 12 shows that the theoretical prediction was in good agreement with the experimental data. In the third experiment, we measured the constraint force of UC gels, discovering that our theory predicted experimental findings under different gel contents with good agreement (Fig. 13). 4.3. Micro mechanical niche around individual fibroblasts The model intuitively reveals unprecedented details about the micro mechanical niche around individual fibroblasts. For FF gels, the traction force exerted by individual fibroblasts reached a maximum between the first and second day of culture (Fig. 14A).
Table 3 Values of the parameters to fit the experimental data of compaction ratio. Collagen
0.5 mg 6
Cells ( 10 )
a
1.0 mg b
Free floating (formula (26)) 0.3 0.78 0.0290 0.9 0.57 0.0088 1.5 0.55 0.0033 Uniaxial constraint (formula (27)) 0.3 e e 0.9 0.10 0.0003 1.5 e e
1.5 mg
a
b
a
b
1.27 0.91 0.83
0.083 0.029 0.022
2.56 1.78 1.38
0.163 0.075 0.043
0.50 0.25 0.20
0.0025 0.0015 0.0013
e 0.80 e
e 0.0015 e
The range of the maximum traction force with the gel contents in this study was 0.62e17.70 nN (0.62 nN occurred in the 0.3 million fibroblasts e 1.5 mg collagen gels, which was not plotted on Fig. 14A); this was well correlated with the ratio of cell number to collagen mass (RCC) (Fig. 14B), in that the maximum traction force was proportional to RCC in the range RCC
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The mechanisms of fibroblast-mediated compaction of collagen gels and the mechanical niche around individual fibroblasts Zhonggang Feng a, *, Yusuke Wagatsuma a, Masato Kikuchi a, Tadashi Kosawada a, Takao Nakamura b, Daisuke Sato b, Nobuyuki Shirasawa c, Tatsuo Kitajima d, Mitsuo Umezu e a
Graduate School of Science and Engineering, Yamagata University, Japan Graduate School of Medical Science, Yamagata University, Japan Department of Anatomy and Structural Science, Yamagata University, Japan d Electronic Systems Engineering, Malaysia-Japan International Institute of Technology, Malaysia e Integrative Bioscience and Biomedical Engineering, Graduate School of Waseda University, Japan b c
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 March 2014 Accepted 24 May 2014 Available online 26 June 2014
Fibroblast-mediated compaction of collagen gels attracts extensive attention in studies of wound healing, cellular fate processes, and regenerative medicine. However, the underlying mechanism and the cellular mechanical niche still remain obscure. This study examines the mechanical behaviour of collagen fibrils during the process of compaction from an alternative perspective on the primary mechanical interaction, providing a new viewpoint on the behaviour of populated fibroblasts. We classify the collagen fibrils into three types e bent, stretched, and adherent e and deduce the respective equations governing the mechanical behaviour of each type; in particular, from a putative principle based on the stationary state of the instantaneous Hamiltonian of the mechanotransduction system, we originally quantify the stretching force exerted on each stretched fibrils. Via careful verification of a structural elementary model based on this classification, we demonstrate a clear physical picture of the compaction process, quantitatively elucidate the panorama of the micro mechanical niche and reveal an intrinsic biphasic relationship between cellular traction force and matrix elasticity. Our results also infer the underlying mechanism of tensional homoeostasis and stress shielding of fibroblasts. With this study, and sequel investigations on the putative principle proposed herein, we anticipate a refocus of the research on cellular mechanobiology, in vitro and in vivo. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Collagen Fibroblast Hydrogel Modelling Cell culture
1. Introduction Fibroblast-mediated compaction of collagen gels attracts widespread attention in studies related to wound healing [1,2], cellular fate processes [3e6], and regenerative medicine [7e11]. Remarkably, however, more than three decades after this phenomenon was first described in detail by E Bell et al. [12], the underlying mechanism and the micro mechanical environment around individual cells still remain obscure. Fibroblasts in newly formed collagen gels are characterised by their striking ability to protrude pseudopodial extensions e as long
* Corresponding author. Graduate School of Science and Engineering, Yamagata University, Johnanmachi 4-3-16, Yonezawa City, Yamagata Prefecture 992-8510, Japan. E-mail address: [email protected] (Z. Feng). http://dx.doi.org/10.1016/j.biomaterials.2014.05.072 0142-9612/© 2014 Elsevier Ltd. All rights reserved.
as several tens of mm within a few hours. These pseudopodia possess a microtubule core and actin-rich tips [13]; their extension has been demonstrated to be integrin mediated [14], and Rhofamily kinases are involved in the regulation of the actin-myosin dynamics of the cytoskeleton [5,15]; and, they ultimately constitute a network via intercellular contacts [16,17]. In this process, collagen fibrils adjacent to the pseudopodia (or in the front of the tips thereof) become aligned with the long axis of the pseudopodia, which induces a local compaction of the network [18,19]. Morphological investigations have generally suggested that this compaction is caused by cells exerting a traction force on the attaching collagen fibrils, which in turn results in the accumulation of fibrils around the cells. However, recent studies have challenged this opinion [20]. At present, the anisotropic biphasic theory (ABT) [21] is the only sophisticated theory on this subject. This approach regards the
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collagen gel as a continuum biphasic material wherein the cells are simply a component of the network phase, which is phenomenologically modelled as a Maxwell-like fluid. The factor that drives gel compaction in the governing equations of ABT is a multiplicative term comprising cell-generated stress, cell concentration, and cell orientation. In other words, ABT explains collagen gel compaction as an integrated effect involving cell traction, cell migration and proliferation, and cellular and fibrillar reorientation. The deeply interactive nature of the various biological and physical elements of the gel compaction process deters the application of a simple cause-and-effect description, and it would be appropriate to approach the problem with due presumption of complexity [22e26]. The micro mechanical environment around individual cells undoubtedly plays a vital role in gel compaction, initially drawing attention due to its close relation to the physiological and pathological behaviours of wound healing [27,28]. And recently, the importance of this issue was raised in studies of the stem cell's niche [29e32]. The force exerted by individual fibroblasts in collagen gels has been measured in the range of 0.1e450 nN by means of direct measurement in gel slabs [33e36] or extrapolation through the deformation of the substrate [37e40]. Although these data show that the cellular contractile force is variable and sensitive to the extracellular environment and culture conditions, the overall micro mechanical environment has not been fully elucidated, quantitatively. Therefore, despite the multitude of studies based on this model tissue, there is still a lack of clear answers to the following important questions. What is the physical process by which cells exert traction force on the surrounding fibrils while compacting the entire construct? While the effects of cellular random migration and fibrillar orientation are spatially limited to the immediate vicinity of the populated cells, the majority of the macro-compaction of the gels is attributable to those fibrils distal to the cells. This begs the question: In what state are those distal fibrils and what roles do they play in the process? A specific value of traction force results from the interplay between cells and the surrounding collagen fibrils. The mystery herein is the underlying regulatory mechanism by which a cell acquires so particular a value from within the broad range of its force-generating capacity. These questions motivate us to try to disentangle the convolution of cause-and-effect in the hope of uncovering the initiating agent of collagen gel compaction, and to expose the micro mechanical niche around individual fibroblasts, by means of mechanical modelling. We particularly focus on the analysis of the collagen network because this, as the passive partner to the active populated cells in the compaction, is much more stable and easily identified. Although the fibril network itself is also a complex structure composed of random cross links and entangled fibrils, it is more readily elucidated than the populated living cells whose behaviour it exactly reflects, as the cell-fibrillar network provide the only primary mechanical action-counteraction coupling within the system [41]. Our approach is based on the statistics of fibrillar networks [42] and is inspired by network alteration theory [43]. We introduce the classification of fibrils into three different types according to their roles during the compaction process and deduce the respective governing equations that exactly describe the mechanical behaviour of each type. In particular, a putative principle based on the stationary state of the instantaneous Hamiltonian of the mechanotransduction system is employed in the deduction. We justify our theory by applying it to two representative fibroblastcollagen gel compactions: free-floating and quasi-uniaxial constraint. In the course of this justification, intriguing findings about the micro mechanical environment around individual fibroblasts in these two systems are obtained. Finally, we discuss the
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implications of our findings in this model tissue with regard to its predicated equivalence to the relevant cellular biology and connective tissue physiology. 2. Experimental materials and methods 2.1. Gel formation and compaction measurement Two types of compactions, free floating (FF) and quasi-uniaxial constraint (UC) (Fig. 1), were experimentally investigated in this study. Modelling was evidenced and verified by the experimental data from these gels, and in turn revealing the (micro) mechanical niche around individual fibroblasts within the gels. Fabrication of the two types of fibroblast-collagen gels was as described in our previous studies [9,10]. Briefly, type I collagen (extracted from rat tail) and dermal fibroblasts (explanted from Wistar rat embryos and subcultured to 5 passages) were mixed with DMEM solutions and poured into two types of casting moulds (Fig. 1A,B). Each gel had a 1.0 ml initial volume; cell number and collagen mass are listed in Table 1. For UC gels, two T-shaped stainless steel wires were initially placed, one anchored to the well edge and the other connected to a vision-based micro force sensor (described below in Measurement of the constraint force in UC gels), for measuring the constraint force during gel compaction. The mixtures were allowed to gelate in a 5%CO2, 37 C incubator for 40 min. After gelation, the FF gels were displaced from the dish bottom by adding culture medium, resulting in a free-floating culture. The ends of the UC gels wrapped spontaneously around the T-shaped anchors. After culture medium was added to float the gels, compaction along the anchor direction was essentially constrained to the region between the two anchors, while compaction could occur freely in the orthogonal directions. As a result, the gels compacted to a string-like shape. The gels were cultured in 6 ml of DMEM supplemented with 1% penicillinestreptomycin and 10% foetal bovine serum (FBS) for one week. Gel size was obtained by image processing of gel pictures using Adobe Photoshop CS4, and compaction was delineated as the ratio of the gel area (for the FF gels) or of the gel width (for the UC gel) to the corresponding initial values. A surfacemarked cover glass was placed on the gels when gel thickness was measured, and the distance between the lower surface of the cover glass and the bottom of the culture dish was measured by means of an inverted light microscope. 2.2. Measurement of the elasticity of FF gels under external compression Gel was immersed in phosphate buffer saline (PBS) and settled on the stage of an analytical balance. A controlled press moved downward to compress the gel at constant speed. After the compression strain reached 5%, the compression was halted and stress relaxation occurred. The compression/relaxation profile was regressed via a model composed of two parallel Maxwell fluid models (details in Supplementary information S1). The elasticity was regarded as owing mainly to the collagen fibrillar network, and was used to verify the theory of fibrillar bending (see 3. Modelling Theory). 2.3. Measurement of the constraint force in UC gels We developed twenty vision-based micro force sensors and individually calibrated them. The principle of these sensors is that transverse force can be investigated by measuring the deflection of glass fibre cantilever through microscope (details in Supplementary information S2). Each sensor was settled onto one UC gel mould (as shown in Fig. 1B) and the constraint force during the gel compaction can be measured. 2.4. Observation of cellular morphology in gels Following the addition of 0.3 ml acetic acid (99.7%, Sigma) to the culture dishes of 1 mg collagen-0.9 million fibroblasts gels (with a confirmed decrease in pH to 3.20) and incubation in a 37 C incubator for 30 min, the collagen gel was dissolved. This allowed the morphology of the embedded fibroblasts to be clearly observed under a biological inverted microscope. Some of the cells that had formed a pseudopodial network within the gel could be separated by gentle pipetting. Pictures were taken of cells with pseudopodial processes, and measurements were made using Adobe Photoshop CS4; these included the diameter of the cellular soma, and the diameter and length of the pseudopodial processes. From these data, cellular surface area could be estimated. 2.5. Observation of collagen fibril morphology by scanning electron microscopy (SEM) SEM samples for FF gel, UC gel, and cell-free gel were prepared by using standard protocol for biological tissues. 2.6. Observation of the expansion of the gel immediately after cellular necrosis This experiment was to demonstrate that the nature of the fibrillar bending as gel compaction is of elastic. Cell necrosis was induced by adding 0.1 ml of 1 N sodium hydroxyl to the FF gel dishes containing 0.9 million cells in 1.0 mg collagen at 8 h culture. Then, gel diameter was measured by means of an inverted microscope.
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Fig. 1. Two types of fibroblast-collagen gels were investigated in this study: (A) a gel under free-floating compaction, FF gel; (B) a gel under uniaxial constraint compaction, UC gel. Traction force during compaction is measured by a vision-based micro force sensor as described below.
2.7. The coefficient of data determination R2 In this study, the coefficient of data determination R2 was defined as P 2 0 0 R2 ¼ 1 N1 N n¼1 ððxn xn Þ=xn Þ ; where xn are the experimental data, xn are the corresponding theoretical regression values, and N is the number of experimental data.
Nevertheless, there remained numerous randomly orientated fibrils among those stressed e many connecting the parallel, stressed fibrils (Fig. 2H). 3.2. Structural elementary model of the compaction
3. Modelling theory 3.1. Evidences of the modelling The structural elementary model of collagen gel compaction developed in this study is supported by the morphological observations on populated fibroblasts and collagen fibrils. Therefore, some of the experimental results related with these observations should be firstly exhibited herein. Fig. 2 shows the evidences of the model from the cellular and fibrillar morphology. Just after 2 h of culture almost all fibroblasts protruded pseudopodial processes to the surrounding collagen fibrils (Fig. 2A). At 4 h, the processes were much longer and cells had gathered closely together (Fig. 2B). At 8 h, the processes contacted each other, forming a 3-dimensional framework (Fig. 2C). From 8 h onwards, most of the cells had merged multiple pseudopodia, assuming a dendritic polar shape (Fig. 2D). In the fibrillar aspect, fibrils in the cell-free gel were relatively straight (Fig. 2E), whereas those in the fibroblast-compacted gel were curved (Fig. 2F) e evidence that almost all fibrils ought to be stretched at the beginning of gel-compaction in case, and finally most fibrils are actually bent as a consequence of gel compaction. The morphology of the fibrillar network in the UC gel was characterised by the linear alignment of the stressed fibrils parallel to the constraint axis (Fig. 2G).
Z
Table 1 Gel compositions used in this study. Collagen:
0.5 mg
ðG þ T þ BÞ dsu 1.0 mg
1.5 mg
Cells ( 106) Free floating (FF) 0.3 0.9 1.5 Uniaxial constraint (UC) 0.3 0.9 1.5 (initial volume ¼ 1.0 ml)
Supported by the above evidences, we abstracted the whole gel into two interacting elementary units (Fig. 3), each of which consists of one central cell and the surrounding collagen fibrillar network. Collagen fibrils are classified into three types: bent (curved lines), stretched (straight lines), and adherent (shadow around the cell). The mechanical state in the horizontal direction is depicted here. T and B are the stretching force and bending force in the corresponding fibrils, respectively. G is the boundary force. In this model, the compaction process begins as the cell protrudes pseudopodia that adhere to adjacent collagen fibrils. Among these adherent fibrils are some with the correct length potential for sustaining the traction force exerted by the two respective pseudopodia, rooted in neighbouring cells. Almost all fibrils have become stretched at the beginning of compaction can be assumed since they are then rather straight (Fig. 2E and verified by the results: Fig. 16B). Due to the existence of a free (i.e., moveable) boundary, the traction force results in a loss of mechanical equilibrium in the direction normal to the boundary; consequently, cells near the boundary are compelled to move closer to those at more interior locations. This movement bends the majority of the fibrillar network. As long as a cell continues protruding or extending its pseudopodia, the loss of mechanical equilibrium would continue until the fibrillar network between the two cells becomes stiff enough to resist the traction force. The equation governing the movement of the elementary unit is then given by Newton's second law,
B B B
B B B
B B B
B
B B B
B
Z
Rc dsc þ Rf dsf
dv ¼ mc þ mf dt
(1)
where: G, T, and B are the boundary force, stretching force, and bending force, respectively; Rc and Rf are the fluid resistance on the cell and the fibrils, respectively; mc and mf are the mass of the cell and the fibrils, respectively; v is the movement velocity, and t is the time. The inertial term on the right side of Eq. (1) is so small as to be negligible. Evaluation of Rc and Rf by using Lamb's formula reveals that they too can be neglected. Consequently, the situation becomes a static problem:
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Fig. 2. Morphological observations of populated fibroblasts by means of acetic acid dissolution (AeD), and of collagen fibrils by means of scanning electron microscopy (EeH). Fibroblasts in FF gel just after 2 h of culture (A), after 4 h of culture (B), after 8 h of culture (C), and cells separated by lightly pipetting from cellular framework after 8 h of culture (D). Fibrillar networks in the cell-free gel (E), in the fibroblast-compacted FF gel (F), and in the UC gel (G, H).
Fig. 3. Structural elementary model of gel compaction.
GþT þB¼0
(2)
3.3. Theory of fibril mechanics in compacted gels In this study, we originally classified the fibrils into three types e bent, stretched, and adherent e and deduced the respective equations governing the mechanical behaviour of each type as below.
3.3.1. Fibril bending force Fig. 4 illustrates the concept of fibril bending during gel compaction. l is the average size of the fibrillar network mesh; ε is the compaction strain; and q is the bending displacement. Note that affine deformation is assumed. Compaction is effected by the bending force B in the medial fragment, which deforms the horizontal fibril, rather than by buckling of the medial fragment itself. Estimation of l is based on the statistical theory of fibre packing developed by S Toll [42], which deals with realistic random fibrillar networks and yields the formula
l¼
Fig. 4. Mechanical schematic for modelling a bent fibril: the compaction displacement, vertical in this schematic, induces bending of the horizontal fibril fragment.
pr 4ff
(3)
where f is the fibril volume fraction, f is a scalar invariant of fibre orientation distribution (equal to p/4 or 2/p for a 3D or 2D random orientation, respectively), and r is the fibril radius. The bending force B is defined as
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B¼
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vE vq
(4)
where E is the bending energy. For the semi flexible collagen fibril [44],
E¼
1 k QA 2 B
I
k2 ds
(5)
where kB is Boltzmann's constant (1.380658 1023 J K1), Q is absolute temperature, A is the persistent length of the collagen fibrils, k is the bending curvature, and s is the contour distance from one end. To calculate the curvature k we adopted network alteration theory [43], which regards the kinematics of a filamentous network as a serially formed structure that may be substantially unlike the original one. By this concept, the compaction of the fibrillar network is stepwise. At each step, the network is reformed so as to acquire new values of the network state parameters f and l. The amount of bending in each step is small (as in the gently curved fibrils shown in Fig. 2F), and can be approximated by a sine curve (in accord with Ref. [45]). Thus,
1 k QA 2 B
E¼
¼
I
1 k QA 2 B
00
0
y
(6)
2
1 þ y02
5=2 dx
where y ¼ q sinðp x=2lÞ ¼ ε l sinðp x=2lÞ is the orthogonal coordinate of the fibril, x is the horizontal coordinate, and primes denote the derivative with respect to x. Substituting y into the above equation and recognising it is a small quantity, we can approximate deformation energy as
1 Ez kB Q A 2
Z2l y
00
5 1 y02 2
dx
¼
p4 kB Q A ε2 32 l
p4 kB Q A ε 2 32 l
Hb ¼
1 k QA 2 B
I k2 ds
(11)
He ¼
1 2
I
Gp r 2 ðs=s0 1Þ2 ds
(12)
where G is the elastic modulus of the fibril, s is the contour distance from one end, and s/s0 represents the elastic extension ratio at s. Hq is the work done by the bending force:
Hq ¼ Bq
(13)
where q is the transverse displacement due to the bending force B. Ht is the traction energy exerted by cells that stretch the fibril at its two ends:
(14)
(7)
Therefore, the average bending force at the current step can be expressed as
B¼
(10)
Hb is the bending energy, again calculated by
Ht ¼ T 2l 2l0
0
p4 kB Q A q2 z 3 32l
H ¼ Hb þ He þ Hq þ Ht
He is the energy of elastic elongation, which can be approximated by
k2 ds Z2l
generate traction force arbitrarily but rather in accord with the surrounding environment. As shown in Fig. 5A, the fibrils are stretched by cells from either end, as in a “tug-of-war”, yet are under the influence of a transverse force e the bending force that is applied by the fragments of the random portion of the fibrillar network connecting the stressed fibrils (as evidenced in Fig. 2H). This bending force, as formulated by Eq. (8), is stored in the random fibril network and provides the environment that surrounds the cell while exerting traction force on the stretched fibrils. We analysed a representative fragment in the stretched fibril, as shown in Fig. 5B, in order to discover the relationship between the traction force and the bending force. The effective Hamiltonian for the fragment may be written as the sum of four terms [49]:
where 2l0 signifies the initial end-to-end length of the fragment. To find the relationship between traction force T and bending force B, we consider the fragment to be initially straight (as evidenced by Fig. 2E), before being bent by force B and stretched by force T. The curve of the fibril is approximated as a sinusoid [50] with amplitude q. Thus the contour length after deformation is
(8)
In addition to computation of the bending force applied to the elementary unit, the above equations also allow estimation of the elastic modulus of the gel (denoted by X) under external compression, during its compaction e as long as ε is equated to the external compression strain and bending force to the external pressure:
X¼
p4 kB Q A 1 2 32 l
(9)
3.3.2. Stretching force Generally speaking, the traction force is generated in the cytoskeleton and transmitted to the stretched fibrils via cellular mechanotransduction [46e48]. The proposed concept for deducing the traction force applied to each stretched fibril is that cells do not
Fig. 5. Depiction of a stretched fibril (A): the fibril is stretched by the traction force exerted by two neighbouring cells under the influence of a transverse bending force, which originates from the random fibrillar network. Free body diagram of a representative fragment within the stretched fibril (B).
Z. Feng et al. / Biomaterials 35 (2014) 8078e8091
S¼
#12 Z2l " pq px 2 p2 q2 cos dx ¼ 2l þ 1þ 8l 2l 2l
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(15)
0
If we assume the elongation within the fibril to be uniform, then
1 2
He ¼
2 s Gp5 r 2 q4 1 ds ¼ 3 s0 256l
I Gp r 2
(16)
since s0 ¼ 2l. However, q cannot be set by the affine condition as was done for the random fibrillar network. Here, we propose an important hypothesis that the cells stretch the fibrils in such a way as to achieve stability of the Hamiltonian, that is
dH¼0
(17)
where dH is the variation of H as the fibril is stretched by D at the ends. Energy conservation during bending leads to
Fig. 6. Alteration of cellular surface area during culture in collagen gel. Squares mark the experimental estimates (n ¼ 10). The solid line is derived from Eq. (23).
Bq ¼ Hb þ He
The adhesion force between the cell and individual adherent fibrils may be evaluated as
¼
kB QAp4 q2 3
32l
þ
Gp5 r 2 q4
(18)
3
Thus, 3
256l B ¼ 8p4 kB QAq þ Gp5 r 2 q3
(19)
As the fragment is stretched by the cell, extending the ends by D, the corresponding change in H is
1 q 1 q3 kB QAp4 3 dq þ Gp5 r 2 3 dq þ TD dH ¼ Bdq þ 16 64 l l
(20)
Since by Eq. (15),
dq 4l z 2 D p q
(21)
Substituting Eq. (21) into Eq. (20) and apply Eq. (17), we obtain
T¼
4l k QAp2 Gp3 r 2 q2 Bþ B 2 þ 2 2 p q 4l 16l
(22)
Thus, combining Eqs. (19) and (22) we can obtain the stretching force sustained in individual collagen fibrils. 3.3.3. Fibrils adherent to the cell and evaluation of the fibrillar adhesion force To estimate the total length of the fibrils adherent to the cell, we measured the change in cellular surface area in the cellular morphology images. Fig. 6 shows that the following formula can satisfactorily fit the measurement data:
Sc ¼ 4p r02
a þ b tegt a þ tegt
(23)
where r0 is the initial radius of the cell, t is the culture duration, and a, b, and g are constants. The adhesion of collagen fibrils is regarded as a two dimensional fibrillar web, modelled as a l l mesh wrapped on the cellular surface. Therefore, the total length of the adherent fibrils may be estimated as
Lad ¼
2Sc l
2
t ¼ Bc
256l
(24)
l Sc
(25)
The total bending force Bc can then be interpreted as the resistance to cell movement, distributed between the adherent fibrils. 3.4. Computation procedure In a practical computation, the first step is to calculate fibrillar mass and initial volume in the elementary unit. Then, from the measurement of gel compaction, the unit volume during compaction can be obtained. At each compaction step, the cellular surface area is computed along with the masses of the adherent fibrils, the stretched fibrils, and the bent fibrils. Following this, Eq. (3) is applied to calculate the average length of the bent fragment l. For the FF compaction, by Eq. (2) traction force is equal and opposite to bending force in any direction, because the force at the boundary is zero. For the UC gels, the traction force is equal to the bending force in the lateral direction with the free boundary, whereas it is equal to (GþB) in the constraint direction. The values of the fundamental constants and parameters used in the computations are listed in Table 2. 4. Results 4.1. Fibroblast-mediated collagen gel compaction Experimental results of gel compaction with different gel contents (Table 1) under free floating and constraint conditions are Table 2 Values of the fundamental constants and parameters in the model. Time interval (h)
r0b (mm)
rb (mm)
Tb (K)
Ac (mm)
Gd (GPa)
ab
bb
gb
1.0a
9.5
0.025
310
120
0.4
50
6.0
0.25
a
Gel compaction is several percent over this time interval, coinciding with the result of the gel expansion experiment. Contacts at the fibrillar intersections are required to be stable over a comparable period. b These values were measured as part of this study. c The persistent length of the collagen fibrils was determined by estimating SEM of gels without cells and referred to Refs. [44,51,52]. d The elastic modulus of the collagen fibrils was determined by referring to Refs. [53e55].
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shown in Figs. 7e10. We found that the following formulae can fit the experimental measurement in good agreement.
J2D ¼
a þ bt aþt
J1D ¼
a þ bt aþt
(26) 12 (27)
where J2D and J1D are the area ratio for free-floating compaction and width ratio for constraint compaction, respectively. a and b, listed in Table 3, are constant with respect to each cell number and initial collagen concentration (ICC); t is the culture duration in hour. Summarily, dramatic compaction occurred within the first 12 h (Figs. 7 and 8). Volume compaction ratio was proportional to ICC, but reverse related with cell number as the other condition keeping constant (Figs. 9 and 10). Compaction of UC gels was consistently more pronounced than that of FF gels (Fig. 10). 4.2. Verification of the model Verification of the structural elementary model was provided via the results from the experiments of gel expansion, elastic measurement, and constraint force measurement. In the first, we observed ca. 3% expansion in the diameter of FF gels after inducing necrosis of populated fibroblasts (Fig. 11). This shows that the fibrillar bending is elastic, which is a prerequisite of our theory to compute bending force and is in keeping with network alteration theory; specifically, this allowed the gel compaction to be analysed as a series of small, stepwise deformations. To demonstrate that the observed expansion was not due the change of pH of the medium caused by addition of sodium hydroxyl, we added acetic acid to the gel culture medium; expansion was again observed, and to the same extent. Particularly notable was that cell-free gels subjected to these conditions underwent contraction rather than expansion. In the second experiment, we measured the elastic modulus of FF gels and compared it with the prediction in our theory by Eq. (9). Fig. 12 shows that the theoretical prediction was in good agreement with the experimental data. In the third experiment, we measured the constraint force of UC gels, discovering that our theory predicted experimental findings under different gel contents with good agreement (Fig. 13). 4.3. Micro mechanical niche around individual fibroblasts The model intuitively reveals unprecedented details about the micro mechanical niche around individual fibroblasts. For FF gels, the traction force exerted by individual fibroblasts reached a maximum between the first and second day of culture (Fig. 14A).
Table 3 Values of the parameters to fit the experimental data of compaction ratio. Collagen
0.5 mg 6
Cells ( 10 )
a
1.0 mg b
Free floating (formula (26)) 0.3 0.78 0.0290 0.9 0.57 0.0088 1.5 0.55 0.0033 Uniaxial constraint (formula (27)) 0.3 e e 0.9 0.10 0.0003 1.5 e e
1.5 mg
a
b
a
b
1.27 0.91 0.83
0.083 0.029 0.022
2.56 1.78 1.38
0.163 0.075 0.043
0.50 0.25 0.20
0.0025 0.0015 0.0013
e 0.80 e
e 0.0015 e
The range of the maximum traction force with the gel contents in this study was 0.62e17.70 nN (0.62 nN occurred in the 0.3 million fibroblasts e 1.5 mg collagen gels, which was not plotted on Fig. 14A); this was well correlated with the ratio of cell number to collagen mass (RCC) (Fig. 14B), in that the maximum traction force was proportional to RCC in the range RCC
