Modelling the Contraction Process of Collagen Gels
Populated with Living Cells
Zhonggang Feng1, Tatsuo Kitajima2, Takao Nakamura3, Nobuyuki Shirasawa4, Mitsuo Umezu5
1

Graduate School of Science and Engineering, Yamagata University, Japan
Electronic Systems Engineering, Malaysia-Japan International Institute of Technology, Malaysia
3
Graduate School of Medical Science, Yamagata University, Japan
4
Department of Anatomy and Structural Science, Yamagata University, Japan
5
Integrative Bioscience and Biomedical Engineering, Graduate School of Waseda University, Japan
2

1

Introduction

Collagen gels will contract as living cells populated withinthem, a process regarded as the in vitro analog of wound healing.
This phenomenon is now harnessed to fabricate engineered
tissues, such as tendons, blood vessels, and cardiac muscle [1].
Hence, to understand the contraction process is quite important
for the studies of wound healing process and for the applications
in tissue engineering. There is no commonly accepted theory
that reveals the behind mechanisms of the contraction. Somemodels include the effects of fibril orientation, fibril network
deformation, cell traction, and cell migration, making thealgorithm mathematically tough and the whole physical image
complicated [2,3]. In this study, we model the early phase of theprocess, postulating that the contraction process is mainly due tothe large scale movement of the populated cells and thesurrounding collagen fibrils, which is driven by the traction
force generated by cells onto their surrounding fibrils. The
movement results in the compaction of fibril network. By meansof this model, we can obtain the volume contraction ratio andthe traction force generated by the cells to the fibrils. The model
is verified by experimental data.

2TB is the bend force; Fr is the interstitial fluid resistance. Eachof these quantities can be calculated as follows,ρ +δ
nT = nc ∫ρ −δ

1−(ρ − ρ )22σ 2(2)dρ ,2π ⋅ σwhere nc is the number of fibrils contacting to the cell; ρ is thelength of fibrils following the Gaussian distribution; ρ and σ
are the mean and standard deviation of ρ, respectively; δ is thetraction tolerance for length ρ.
(3)n B = k ( n0 − nT )k ∈ (0,1) ,where n0 is the total number of the fibril; k is the bendcoefficient.
eTB = k BTAγ 2 / 2 x ,(4)where kB is Boltzmann constant; T is absolute temperature; A is
the persistence length of fibrils; γ is bend curvature; x is the celldisplacement.
1
Fr = dSDv 2 ,(5)2where d is the fluid density; S is the area projected at flow
direction; D is resistance coefficient; v is the movement velocity.

TheoryBased on the observation by means of scanning electron
microscopy, we model the process of early contraction phase as
follows. The gel is regarded as homogeneous fibril network
uniformly embedded with living cells. Therefore, it can bedivided into contraction subunits, each of them including twocells and the surrounding collagen fibrils as shown in Fig. 1.
Due to the interaction between the cells and the fibrils mediatedwith the integrin receptor in the cellular membrane, cells
generate traction force onto their surrounding fibrils. However,
only the traction force exerting to the fibrils that are straight inbetween the cells (named traction fibrils) is sustained, whichconsequently draws the cells with the surrounding fibrils tomove close to each other. The experimental measurement
showed that the length of the fibrils followed the Gaussian
distribution. Therefore, as the two cells moving close theywould encounter more new traction fibrils with shorter lengthand generate traction force to them to pull the cells close further;in the meanwhile, the former longer traction fibrils are out oftension and bent to turn into resistance to the cell movement. In
this process, the moving of the cells and fibrils is also resistedby the interstitial fluid. The motion equation for the mass
including one cell and its surrounding fibrils isa ⋅ Θ = nT τ − nBTB − Fr,(1)
where a is the acceleration; Θ is the mass including the cell andthe surrounding fibrils; nT is the number of traction fibrils; τ isthe traction force for one fibril; nB is the number of fibrils bent;

Fig. 1. Model of collagen gel contraction
The contraction process governed by Equation 1 should be like
this: at the beginning of the contraction, TB and Fr are zero
because no fibrils are bent and movement has not yet occurred;drawing force nTτ produces acceleration and results in the startof contraction. Once the contraction occurs, TB and Fr will
emerge to resist the movement so that the contraction will
undergo at a certain velocity at which drawing force nTτ andresistance TB and Fr get equilibrium. As contraction continues,
the drawing force will increase because the number of traction
fibril nT increases, so that the equilibrium velocity will alsoincreases. The contraction process will halt as the bend force
nBTB is large enough to equate drawing force nTτ at that time.

3Results

The simulation result of the contraction process of a gel
consisting of 1 mg type I collagen and 1.0 million rat fibroblastsis plotted in Fig. 2. The parameters in the simulation arepresented in Table 1. The setting of each parameter is based on
experimental measurement of the gel. It can be seen that thismodel is able to reproduce the contraction process in good
agreement with the experimental data.

the experimental data are in good agreement with each other.
The process does not include cell migration, which may imply
that at least at the early contraction phase cell migration is not
an important factor to the phenomenon. This point is different
from the wound healing process in vivo.
The simulation is only conducted for free contraction, how
to simulate the contraction under external constraint in the same
principle should be further studied.

5Conclusion

The simulation shows that the fundamentals of thecontraction process, at least for the early phase, is that thetraction force exerted by cells is sustained by some traction
fibrils in proper length, which in turn draws the cells and thesurrounding fibrils move close resulting in macro compaction.
Cell migration is not an important factor to this phase.

Acknowledgements
This study is supported in part by a Grant-in-Aid for
Scientific Research (C) from the Japan Society for thePromotion of Science (JSPS) (22500424).
References
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[2]

[3]

Fig. 2. Comparison of the simulation and the experiment
Table 1. Values of fundamental parameters in the simulation
ρ (µm) σ(µm) δ(µm)τ(pN)
kA(µm)
n024000
140.0
5.00.1
6.0
0.2
20.0
Fig. 3. shows the drawing force exerted by one cell to thetraction fibrils. This force is difficult to be measured byexperiment, but it is of interest because more and more studies
in cellular biology evidence that the mechanical stress is a vitalcue to the cellular fates including stem cell differentiation [4].

Fig. 3. Drawing force exerted by one cell to the traction fibrils

4Discussion
This model is able to simulate the early free contraction
process of collagen gels populated with living cells. The
simulation shows that the fundamentals of the contraction
process, at least for the early phase, is that the traction force
exerted by cells is sustained by some traction fibrils in proper
length, which in turn draws the cells and the surrounding fibrils
move close resulting in macro compaction. The simulation and[4]Z Feng, D Seya, T Kitajima, T Kosawada, T Nakamura, M Umezu.
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