Fisher John P. e.a. (ed.) Tissue Engineering

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Cell Migration

Gang Cheng

Kyriacos Zygourakis

Rice University

6.1 Introduction.............................................. 6-1

6.2 Characteristics of Mammalian Cell Migration ........ 6-2

Cell Movement Cycle • Persistent Random Walk •

Cell–Cell Contacts

6.3 Regulation of Cell Movement ........................... 6-4

Soluble Factors Modulate Cell Movement • ECM Proteins

and Cell–Substrate Interactions Regulate Cell Movement •

Electrical Fields Direct Cell Movement

6.4 Cell Migration Assays.................................... 6-7

Cell-Population Assays • Individual-Cell Assays

6.5 Mathematical Models for Cell Migration and Tissue

Growth.................................................... 6-10

References ....................................................... 6-12

6.1 Introduction

Cell migration is an essential component of normal development, inﬂammation, tissue repair, angiogen-

esis, and tumor invasion. After conception, selected cells of the developing mammalian zygote invade the

uterine wall to establish the placenta, while the intricately programmed migration of other cells within

the embryo shapes the complex form of the emerging organism [1,2]. The nervous system is another

example of large-scale cell migration during fetal development. The growth of axons and dendrites is

preceded by a phase of cell migration in which immature neurons (or neuroblasts) move from their birth-

place to settle in some other location in order to make the right connections [3]. Certain kinds of white

blood cells are able to migrate through the walls of blood vessels and into the surrounding tissues, actively

seeking and engulﬁng sources of decay [4]. Migrating ﬁbroblastic and epithelial cells heal wounds, and

osteoclasts and osteoblasts are in constant movement as they remodel bone [5–7]. Tumor cell motility

is also required for invasion and metastasis. The crawling malignant tumor cells that invade and disrupt

tissue architecture account as much or more for the lethality of cancer as does uncontrolled growth [8].

Cell migration also plays a key role in determining the structure and growth rate of bioartiﬁcial tissues

built on scaffolds made from suitable biomaterials [9]. In recent years, a lot of attention has been focused

on the development of biomimetic materials capable of promoting cell functions, including migration,

by biomolecular recognition [10]. Such recognition can be achieved by surface or bulk modiﬁcation

of the material with bioactive molecules such as extracellular matrix (ECM) proteins or short peptide

sequences that can induce speciﬁc interactions with cell receptors. In order to design biomimetic scaffolds

with optimal properties for each application, however, we must thoroughly understand not only the

mechanism of cell migration, but also the many factors that modulate this important process. We must

6-1

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6-2 Tissue Engineering

also develop assays to accurately characterize cell movement on various biomaterials and, ultimately, build

theoretical models that can quantitatively predict the effect of system parameters (scaffold properties,

nutrient or growth factor concentrations, pH, etc.) on tissue development. Because tissues are highly

heterogeneous systems exhibiting complicated cell population dynamics, such theoretical models must be

able to accurately describe cell–cell and cell–substrate interactions.

This review attempts to address some of these issues for anchorage-dependent mammalian cells. After

a brief description of the mechanism of cell movement, we outline the role that growth factors, substrate-

adhesion molecules, and other environmental factors play in modulating cell migration. Since accurate

measurements are essential for elucidating the effect of a speciﬁc stimulus on cell migration, we discuss

the application of several assays that may be used to characterize cell motility. Finally, the use of theoretical

models for analyzing cell population dynamics and predicting tissue growth rates is discussed.

6.2 Characteristics of Mammalian Cell Migration

6.2.1 Cell Movement Cycle

The movement of a mammalian cell on a substrate is an intricate process requiring at least three structural

elements: an ECM ligand on the substrate, its cell surface receptor, and the intracellular cytoskeleton [6].

The receptors that play key roles for cell movement belong to a large family of transmembrane proteins

called integrins [11]. Migration can be considered as a continual cycle consisting of four essential steps

[4] (1) extension of the cell’s leading margin over the substratum to form a lamellipod (i.e., a thin piece

of membrane and cytoplasm at the front of the cell); (2) attachment to the substrate; (3) pulling or

contraction using the newly formed points of adhesion as anchorage; and (4) release or detachment of

adhesions at the rear of the cell. These four steps are orchestrated by the interaction of various extracellular

and intracellular molecules. Extension of the leading margin of the cell is caused by the polymerization

of actin ﬁlaments at the lamellipodia and crucial factors involved in this process are the Arp2/3 complex,

gelsolin, and capping protein [4,12,13]. At the base of the cortical actin meshwork, coﬁlin promotes

the disassembly of ﬁlaments [4,14]. Integrins anchor the cell to its substratum by binding both to ECM

molecules on the outside of the cell and to the actin cytoskeleton on the inside. Cortical contractions

due to myosin molecules pull structures toward the center of the cell, causing the uropod (lamellipod’s

counterpart at the rear of the cell) to retract and unattached structures on the dorsal surface to move

backward [15–17]. The cycle is completed by a forward movement of actin and other constituents through

the cytoplasm to the leading margin of the cell [4].

6.2.2 Persistent Random Walk

Migration of individual mammalian cells in isotropic environments can be described as a persistent

random walk [18,19]. Over short time periods, cells follow a relatively straight path, showing persistence

of movement (see Figure 6.1a). If long time intervals are used to observe the cell position, however, cell

movement appears similar to Brownian motion with frequent direction changes (Figure 6.1b). At least

two parameters are needed to describe persistent random walk [20]. The ﬁrst one is the speed S that is

intuitively deﬁned as the displacement of the cell centroid per unit time. The second one is the persistence

time (usually denoted by P) that is a measure of the average time between “signiﬁcant” direction changes.

The magnitude of P and S depend both on the type of the cell and on its microenvironment. Reported

values of migration speed and persistence time range from 0.5 µm/min and 4 to 5 h, respectively for

human microvessel endothelial cells and smooth muscle cells [20,21] to 20 µm/min and 4 min for

rabbit neutrophils [22]. Lauffenburger and coworkers [23] pointed out that there exists a rough inverse

relationship between S and P and that this could be understood by considering the product of these two

parameters as the cell’s analog to a “mean free-path length.” Rigorous deﬁnition of P and S necessary for

mathematical modeling and the assays to measure them will be described in Section 6.4.

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Cell Migration 6-3

200

300

400

900 1000 1100 1200 1300

Start

200

300

400

900 1000 1100 1200 1300

Start

Observation interval= 3 h

Observation interval= 0.5 h(a)

(b)

X, mm

Y, mm

FIGURE 6.1 Typical trajectory of a bovine pulmonary artery endothelial cell migrating in a uniform environment.

Symbols represent the position of the centroid of the same cell recorded at 30 min intervals (top panel) and 3 h

intervals (bottom panel). When the observation interval is short (top panel), the cell clearly exhibits persistence in

movement direction. If a long observation interval (bottom panel) is chosen, however, the movement of the same cell

appears to be a random walk with frequent direction changes. (Adapted from Lee, Y., Mcintire, L.V., and Zygourakis,

K., Biochem. Cell Biol., 1995, 73: 461–472. With permission.)

6.2.3 Cell–Cell Contacts

In a population of migrating cells, the persistent random walk of individual cells can be interrupted by

contacts with other cells. Since cells usually stop and change the direction of their movement after such

contacts, this phenomenon is often called contact inhibition of locomotion [6]. When two ﬁbroblasts collide

with each other, for example, rufﬂing of the membrane near the contact point stops to form a quiescent

region, while rufﬂing continues at other regions of the membrane. After about 25 min, the cells break

the adhesion and move away in new directions [6]. Cell–cell contacts have an even more profound effect

on the migration of epithelial cells. Following a collision, the leading lamellae of the epithelial cells are

gradually lost and an adhesion is formed between two cells. Sequential collisions with other cells result in

the formation of small colonies and eventually a sheet of contiguous cells [6,4]. This process is essential

for the coverage of the wounded area in wound healing [24,25]. It should be noted that cell–cell contacts

also affect cell division and the effect is also often referred to as “contact inhibition,” which actually

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6-4 Tissue Engineering

means contact inhibition of division. As is shown later, the mechanisms of these two contact inhibitions

are different, even though they are related in a rather complicated way.

6.3 Regulation of Cell Movement

6.3.1 Soluble Factors Modulate Cell Movement

Many polypeptide growth factors can upregulate cell motility. Sato and Rifkin [26] found that when a

conﬂuent monolayer of bovine aortic endothelial cells (BAECs) was wounded with a razor blade, cells

at the edge of the wound released basic ﬁbroblast growth factor (bFGF), which stimulated the rapid

movement of nearby cells into the denuded area. Addition of anti-bFGF IgG slowed down considerably

the cell movement and this inhibition was dose dependent. Sato and Rifkin also found that bFGF regulated

the basal level of synthesis of the protease plasminogen activator (PA) and the basal level of DNA synthesis.

These ﬁndings are consistent with earlier work demonstrating that plasmin contributed to cell migration

[27,28]. Sato and Rifkin [26] also offer the alternative explanation that bFGF acts as a motility factor

via its adhesive interactions described by Baird and coworkers [29]. Platelet-derived growth factor-BB

(PDGF-BB) has been found to be the major motility enhancing factor in human serum for human dermal

ﬁbroblasts migrating on type I collagen, even though it is not needed for the initiation of cell movement

[30]. Transforming growth factor-α (TGF-α) enhances locomotion of cultured human keratinocytes [31].

Stimulation of MTLn3 cells (a metastatic carcinoma cell line) with epidermal growth factor (EGF) causes

rapid and transient lamellipod protrusion along with an increase in actin polymerization at the leading

edge [32]. EGF was also found to greatly enhance random dispersion of ﬁbroblasts by increasing the

frequency of direction changes and at the same time slightly increasing the path length [33]. When

multiple types of growth factors are present, they may affect cell motility synergistically with certain

extent of speciﬁcity. For example, TGF-β1 has been found to synergistically enhance EGF-stimulated

hepatocyte motility responses on collagen-containing extracellular matrices. However, the same effect

was not achieved when TGF-β1 was added together with hepatocyte growth factor (HGF) [34]. Placental

growth factor (PlGF) can augment the migration of endothelial cell in response to vascular endothelial

growth factor (VEGF) in pathological angiogenesis [35].

Recent studies have revealed additional soluble motility-stimulating proteins, which include (a) The

scatter factor or SF (also known as HGF), a mesenchymal cell-derived protein, which causes contigu-

ous sheets of epithelium to separate into individual cells and stimulates the migration of epithelial as

well as vascular endothelial cells [36–38]; (b) the autocrine motility factor (AMF), a tumor cell-derived

protein, which stimulates migration of the producer cells [39–41]; and (c) the migration-stimulating

factor (MSF), a protein produced by fetal and cancer patient ﬁbroblasts, which stimulates penetration of

three-dimensional collagen gels by nonproducing adult ﬁbroblasts [42–44].

The direction of cell movement can be affected by the concentration gradient of certain growth factors,

a response called chemotaxis. Chemotaxis is very important for processes such as normal development,

inﬂammation, angiogenesis, and wound healing in which directed migration of speciﬁc cell types is essen-

tial. It has been found that when a wound is caused in the human body, large amounts of PDGF, EGF,

and TGF-β are secreted at different times by cells around the wound to coordinate the inﬂux of neutro-

phils, macrophages, ﬁbroblasts, smooth muscle cells, and endothelial cells for fast and complete healing

of the wound [45–48]. Lack of these chemotactic growth factors may impair the healing process, while

overproduction can cause excessive repair or scarring [49]. A chemotactic response is typically a func-

tion of both the absolute concentration of the attractant and the steepness of its concentration gradient.

Directional orientation bias increases with concentration gradient steepness, asymptotically approach-

ing a maximal level as steepness increases. The dependence on attractant concentration is biphasic,

increasing at low concentration to reach a maximum, then decreasing as concentration increases further

[50–52].

Several hypotheses have been proposed to explain the underlying mechanisms of these phenomena.

Lackie [6] suggested that cells decide their movement direction by receptor-mediated comparison of

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Cell Migration 6-5

the spatial difference in attractant concentration across the cell dimension. This explains the biphasic

dependence of directional orientation bias on attractant concentration for constant concentration

gradient. At low-attractant concentrations, very few receptors are bound and, thus, only a small ori-

entation bias results. At high-attractant concentrations almost all receptors are bound and, again, only

a small bias is observed. Maximum bias is found for an intermediate attractant concentration, where

the number of bound receptors is signiﬁcant and most sensitive to differences in local attractant con-

centration. It should be noted that during the course of directed cell movement in chemotaxis, there

are periods during which the cell may randomly stray toward the lower concentration or any other

direction. Mechanistic models aimed at simulating chemotaxis must account for these random ﬂuctu-

ations in the cell’s orientation even at the presence of the concentration gradient of a chemoattractant.

Tranquillo and coworkers [53,54] hypothesized that this is caused primarily by the probabilistic kinetics of

receptor/attractant binding. Their model divides a cell into two compartments along its polarization axis

and assumes that the instantaneous numbers of receptor/attractant complexes at the compartment sur-

faces are governed by a stochastic differential equation with both a deterministic and a probabilistic part,

which accounts for the random ﬂuctuations in the binding process. The numbers of receptor/attractant

complexes are then used to calculate the concentration of motile effectors in each intracellular compart-

ment. Finally, the model postulates that the cell changes direction with an angular rate proportional to

the imbalance between the levels of motile effectors in the two compartments. Model results were shown

to provide good prediction for the chemotaxis of neutrophil leukocytes [55]. A recent extension of this

model [56] relaxes several simplifying assumptions regarding receptor dynamics in the original model

using newly obtained knowledge on transient G-protein signaling, cytoskeletal association, and receptor

internalization and recycling, including statistical ﬂuctuations in the numbers of receptors among the

various states.

6.3.2 ECM Proteins and Cell–Substrate Interactions Regulate Cell

Movement

As described earlier, each of the four phases in cell-movement cycle involves the interaction of cell-surface

receptors with the ECM components on substratum surface. The ECM is a molecular complex whose

components include collagens, glycoproteins, hyaluronic acid, proteoglycans, glycosaminoglycans, and

elastins [57,58]. In addition, ECM harbors molecules such as growth factors, cytokines, matrix-degrading

enzymes, and their inhibitors [57]. The distribution of these molecules varies from tissue to tissue and

changes with time during tissue development, making the ECM a highly dynamic system [59,60]. The

binding of ECM molecules to the extracellular domains of integrins triggers the receptor/ligand binding,

trafﬁcking and signaling cascade, activation of transcription factor and expression of target genes, and

eventually results in the regulation of speciﬁc cell functions, which may include adhesion, migration,

proliferation, and differentiation.

Integrin receptors are heterodimeric proteins composed of α and β subunits [11]. At least 15 α and β

subunits have been identiﬁed so far and they pair with each other in a variety of combinations, giving rise

to speciﬁc recognition on the ECM molecules with different selectivity. These combinations include the

ﬁbronectin receptor, α

, α

, and the vitronectin receptor α

[61–63]. While the α

integrin

binds exclusively to ﬁbronectin, α

can bind either to laminin or to collagen-IV and the α

receptor

recognizes ﬁbrinogen, vitronectin, and probably ﬁbronectin. The α

integrin has also been found to

mediate cell motility on ﬁbronectin and vascular adhesion molecule-1 (VCAM-1) independently of the

[64].

The ability of integrin receptors to recognize and bind the short peptide sequences corresponding to

the adhesive domains of ECM proteins stimulated a lot of interest in developing biomimetic materials.

Several studies by Hubbell and coworkers [65–71] have shown that covalent immobilization of adhesive

peptides like RGD or YIGSR (which are the adhesive domains of ﬁbronectin and laminin respectively) on

the surface of glass or polymeric substrates can promote adhesion of endothelial cells. Kouvroukoglou and

coworkers [72] found that such surface modiﬁcations signiﬁcantly enhanced the migration of endothelial

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6-6 Tissue Engineering

cells, a fact that might lead to higher endothelization rates of the surfaces of implantable biomaterials.

Yang and coworkers [73] found that surface modiﬁcation of poly(lactic acid) (PLA) ﬁlms and poly(lactic-

co-/glycolic acid) (PLGA) porous structures with RGD peptides or ﬁbronectin greatly promoted human

osteoprogenitor adhesion, migration, growth, and differentiation. Shin and coworkers [74] found that

bulk modiﬁcation of oligo(poly[ethylene glycol] fumarate) (OPF) hydrogel with a rat osteopontin-derived

peptide (ODP) or a RGD peptide could increase the motility of marrow stromal osteoblasts and accelerate

the expansion of megacolonies of these cells on the surface of the hydrogel.

The sudden introduction of a migratory ECM ligand into the environment of a cell may provide

a key signal for the initiation of cell migration [75]. Similarly, the biosynthetic induction of such a

migration protein or of its receptors might also promote migration [64,75–79]. The driving mechanism

for migration appears to be provided by the physical interactions between speciﬁc sequences in adhesion

proteinsand their receptors [75], and by intracellular contractile proteins[4,17]. The signals involved in the

termination of cell migration include the reacquisition of cell–cell adhesion molecules (e.g., N -cadherin),

often accompanied by differentiation into the ﬁnal tissue type [75].

The speed of cell movement on ECM proteins is regulated not only by the type of receptor/ligand

interactions, but also by the ligand density on the substrate and the ligand afﬁnity for cell adhesion

receptors [80–83]. The complexity of these interactions leads to a biphasic dependence of cell speed on

substrate adhesiveness. Goodman and coworkers [84] found that the migration speed of murine myoblasts

on substrates covered with laminin or laminin fragment E8 depended in a biphasic fashion on the density

of surface ligand. Maximum cell speeds were observed for ligand densities ranging from one third to

one tenth of those required for strongest cell attachment. Low cell speeds were measured when the cells

adhered very strongly or very weakly to the surface. A slow, monotonic increase of cell speed was observed,

however, when the density of ﬁbronectin ligands was increased. DiMilla and coworkers [22] measured the

speed of human smooth muscle cells on ﬁbronectin and collagen-IV. They found that cell speed varied

in a biphasic fashion with increasing cell adhesion strengths for both ligands. Similar results are seen by

varying the number of receptors on the cell surface. Keely and coworkers [85] found that the strength of

adhesion of human breast carcinoma T47D cells to collagen I and IV decreased with decreasing levels of

expression of the α

integrin (a collagen and laminin receptor). T47D clones that exhibited intermediate

levels of adhesion to collagen had the highest motility across collagen-coated ﬁlters, suggesting that an

intermediate density of cell-surface α

integrin optimally supports cell motility.

The reasons for this contrasting behavior were revealed by an elegant mathematical analysis of the cell-

migration cycle by Lauffenburger [86] and DiMilla and coworkers [87]. The ﬁrst key system parameter for

explaining the experimental data is the substratum adhesiveness that is proportional to the surface ligand

density and inversely proportional to the receptor/ligand equilibrium dissociation constant. Substrate

adhesiveness, however, may not only depend on the strength of receptor/ligand association. A later ana-

lysis of Ward and Hammer [88] showed that the formation of focal contacts (which were modeled as

cytoplasmic nucleation centers binding adhesion receptors) and the elastic rigidity of cytoskeletal connec-

tions may signiﬁcantly affect cellular adhesive strength. The second parameter describes the asymmetry in

bond afﬁnity as the ratio of the dissociation rate constants between the front and the rear of the migrating

cell. The model correctly predicted a biphasic dependence of cell speed on substrate adhesiveness, with cell

locomotion occurring over an intermediate and (in many cases) limited range of adhesiveness. The size of

this range was primarily governed by the asymmetry in adhesiveness between the front and the rear of the

cell. Several mechanisms can provide a front-to-rear asymmetry in cell/substrate adhesiveness including

dynamic integrin/cytoskeletal interactions [89] and polarized distributions of integrins maintained by

receptor trafﬁcking mechanisms [90].

The substratum adhesiveness can also affect the direction of cell movement, a response called hapto-

taxis. For example, cells cultured on a surface will move onto tracks of artiﬁcial adhesive material, such as

polylysine or silicon oxide [4]. When ﬁbroblasts are plated on a surface coated with a uniformly increasing

gradient of a charged substance, they turn and move in the direction of increasing adhesiveness [91].

Dickinson and Tranquillo [92] have developed a stochastic mathematical model based on receptor/ligand

binding and trafﬁcking mechanism to provide a mechanistic understanding of how the magnitude and

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Cell Migration 6-7

distribution of adhesion ligands in the substratum inﬂuence cell movement. Additional sources for dir-

ectionality in movement may be simple population pressure, tissue or physical barriers [4,93], or the

three-dimensional structure of the matrix. In a process known as contact guidance, matrix ﬁbers can be

spatially arranged as to facilitate cell movement in a preferred direction [94]. Micromachined grooves cut

into the substratum have similar effect [95].

While properties of the substratum can greatly inﬂuence cell movement, cells can also direct their

own migration by modifying the physical properties of the substratum. For example, pioneering cells

within a population of migrating neural crest cells may be laying down extracellular cues that trailing cells

recognize [96]. When a suspension of human keratinocytes is plated on a ﬁbrin matrix, single cells invade

the matrix and progress through it by dissolving the ﬁbrin and thereby creating tunnels [97].

6.3.3 Electrical Fields Direct Cell Movement

The migration of many mammalian cells is affected by the presence of electric ﬁelds, a phenomenon

called galvanotaxis. Nerve cells can detect electric ﬁeld as weak as 10 mV/mm and turn their growth

cones to move in the direction of the negative pole [4]. Fibroblasts and cells from the neural crest also

move toward the negative pole in a steady electric ﬁeld [98,99]. During wound healing in vertebrates, a

steady lateral electric ﬁeld of 40 to 200 mV/mm is generated in the disrupted epithelia layers to coordinate

the directed migration of epithelial cells from the nearby regions [100]. Some experiments indicate that

when the electric ﬁeld is removed, the wound healing rate is 25% slower, while nearly every clinical trial

using electric ﬁelds to stimulate healing in mammalian wounds reports a signiﬁcant increase in the rate

of healing from 13 to 50% [101].

Cell’s response to electrical ﬁelds typically requires Ca

inﬂux [102], the presence of speciﬁc growth

factors [100] and intracellular kinase activity. Protein kinase C is required by neural crest cells [103]

and cAMP-dependent protein kinase is used in keratinocytes [104], while mitogen-activated protein

kinase is required by corneal epithelial cells [105]. Speciﬁcally, Zhao and coworkers [100] discovered that

corneal epithelial cells cultured in serum-free medium showed no reorientation in an electric ﬁeld until

250 mV/mm and addition of EGF, bFGF, or TGF-β 1 singly or in combination signiﬁcantly restored the

cathodal reorientation response at low ﬁeld strengths. Interestingly, however, the directed migration of

two ﬁbroblastic cells, NIH 3T3 and SV101, was found to be calcium independent and was, instead,

related to the lateral redistribution of plasma membrane glycoproteins involved in cell–substratum

adhesion [98].

6.4 Cell Migration Assays

6.4.1 Cell-Population Assays

The methods used to assess the locomotory capabilities and characteristics of cells fall into two major

categories. The ﬁrst category includes techniques that monitor large populations of migrating cells and

analyze the number density proﬁles of the population after a given time period of migration. The Boyden

chamber is the most popular assay in this category [106–111]. According to this technique, cells are placed

on the upper surface of a micropore ﬁlter installed in the chamber and incubated for a sufﬁcient period of

time to allow the cells to migrate to the lower surface of the ﬁlter. Cell motility is then quantiﬁed either by

counting the number of cells that have migrated through the ﬁlter or by measuring the distance traveled

into the ﬁlter by several of the fastest moving cells. A comparison of the number of cells that passed

through the ﬁlter or of the migration distance measured for various experimental conditions reveal if, for

example, a test substance is chemotactic for the cells under study. With appropriate modiﬁcations, the

Boyden chamber can also be used to study cell migration under ﬂow conditions or temporal chemotactic

gradients [112,113].

Another cell-population assay measures the migration distance of endothelial cells or osteoblasts

released from growth arrest using a silicon or steel template compartmentalization technique [74,114].

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6-8 Tissue Engineering

This assay system employs tissue culture dishes that are subdivided (using, for example, stainless steel

annular rings) into two separate compartments: an inner circular core and an annular outer ring. Cells

are seeded into the inner compartment and grown to conﬂuence. The inner ring is then removed, and

cells (released from growth arrest) start migrating from a sharp starting line. Cell motility is quantiﬁed by

measuring the distance of the migrating front from the starting line. By using a growth-arrested mono-

layer as a starting cell population for the quantiﬁcation of migration, this assay system produces few or no

wounded cells at the migration front. It also allows us to evaluate the migratory response to other cell types

by coculturing them with the test cells after having seeded the effector cells into the outer compartment of

the assay system. Since migration of endothelial cells is one of the critical features of wound repair, several

researchers have also studied the movement of cells from a wound edge into a denuded area formed by

scraping a conﬂuent monolayer with a razor blade [26,115,116]. Migration is then quantiﬁed by counting

the number of cells in successive 125-µm sections from the wound edge.

A third popular technique is the under-agarose assay where cells are allowed to migrate under a layer

of agarose gel deposited on a glass or plastic surface [117–120]. For random motility experiments, a

migration stimulus is incorporated at uniform concentration into the gel and the cell well medium.

For chemotaxis experiments, the migration stimulus is placed in a separate well from which it forms a

concentration gradient by diffusing through the gel. Typically measured quantities are again the location

of the leading front or the total number of cells migrating away from the well.

However, intrinsic cell locomotory properties are not the only factors inﬂuencing the measurements

obtained by cell-population assays. Parameters like the assay chamber geometry and size, initial cell num-

ber, or attractant diffusivity can signiﬁcantly affect the population dispersal and complicate comparison

of different sets of experimental data. To alleviate these problems, a phenomenological model has been

proposed by Keller and Segel [121] and reformulated by Rivero and coworkers [122]. This mathematical

model takes the form of a partial differential equation describing the temporal evolution of the cell number

density proﬁle and having two key parameters (a) the random motility coefﬁcient µ and (b) a chemotaxis

coefﬁcient χ . By comparing model predictions for cell number density proﬁles to experimental proﬁles

measured with the linear under-agarose assay [123] or the ﬁlter assay [124], the random motility µ and

the chemotaxis coefﬁcient χ can be determined. A different approach was recently followed by Cheng

and coworkers [125] to analyze the differential effect of cell migration and proliferation on the expan-

sion of megacolonies of marrow stromal cells cultured on biomimetic hydrogels modiﬁed with RGD or

osteopontin-derived peptides [74]. This study used a discrete model based on the Markov chain approach

to simulate the cell-population dynamics of expanding megacolonies. A comparison of model predictions

to experimental data showed that surface modiﬁcations enhance the expansion rates of cell megacolonies

by upregulating the speeds of cell migration.

6.4.2 Individual-Cell Assays

Cell-population assays cannot provide detailed information on how cells move. They cannot directly

quantify important locomotory parameters (like cell speed, persistence, turn angles, etc.) or evaluate

the effects of external stimuli on these parameters. An accurate characterization of cell locomotion can

only be obtained by continuously monitoring a sufﬁciently large population of migrating cells using a

video microscopy system with digital or analog time-lapse capabilities. In most applications, cells migrate

on two-dimensional surfaces beneath liquid culture media or agarose gels [21,72,126–129]. Procedures

allowing tracking in three-dimensional collagen matrices have also been reported [50,130–136].

By analyzing a sequence of images obtained at ﬁxed time intervals, the actual cell positions at the

corresponding times can be identiﬁed to reconstruct the individual cell trajectories (see Figure 6.1a). The

mean square displacement D

of the cells from their original positions can then be calculated and plotted

vs. time. If cell movement were a random walk, the mean square displacement D

would vary with time

according to

D

=2nµt (6.1)

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Cell Migration 6-9

where µ is the random motility coefﬁcient (formally equivalent to a diffusion coefﬁcient) and n is a constant

depending on the dimensionality of the random walk. For two-dimensional walks n = 2, while for three-

dimensional walks n = 3 [18,137,138]. Equation 6.1 implies that the average distance traveled by a cell is

proportional to the square root of the elapsed time.

As we mentioned in Section 6.2.2, however, mammalian cells migrating in isotropic environments

execute persistent random walks. Dunn [19] and Othmer and coworkers [139] developed the following

mathematical model to describe persistent random walks:

D

=nS

[Pt − P

(1 −e

−t/P

)] (6.2)

where D

 is the mean square displacement of the tracked cells, S is the root-mean-square cell speed, P

is the persistence time, and n is the constant that gives the dimensionality of the persistent random walk

(n is 2 or 3 for 2D or 3D walks, respectively). This model assumes that S and P are time invariant. The

root-mean-square cell speed S and persistence time P can be computed by ﬁtting the experimental D



vs. time data with the persistent random walk model of Equation 6.2. Nonlinear parameter estimation

algorithms [123] must be used, since graphical techniques [19] lead to results dependent on the size of

the time interval. Note that for long times (t  P), Equation 6.2 reduces to the much simpler expression:

D

=nS

Pt (6.3)

Equation 6.3 implies random walk behavior and allows us to compute the random motility coefﬁcient µ as

follows:

µ =

P (6.4)

Many investigators have successfully used the persistent random walk model to quantify cell migration

[72,140–144]. The chemotactic motion of cells in response to a chemical concentration gradient can also

be modeled by adding a directional bias to the persistent random walk process [21]. The magnitude of

the directional bias is characterized by the chemotactic responsiveness κ. By using experimentally measured

values of S, P, and κ, Stokes and Lauffenburger generated computer simulations of theoretical individual

cell paths, which were useful in elucidating the role of cell migration in physiological processes [21].

Although the persistent random walk model has been very successful in assaying and comparing the

motility of cells under various conditions, it cannot provide all the parameters that may be necessary for

an accurate quantiﬁcation of the cell-migration process. Experimental observations with endothelial cells

[127] have revealed that cells slow down or even stop migrating when they divide or when they collide with

other cells. Migrating cells may also stop for a while before changing their direction of movement [127].

A detailed description of cell movement (suitable, e.g., for the computer implementation of migration-

proliferation models of tissue growth [145]) requires the following information (1) the speed of cell

locomotion (swimming speed); (2) the expected duration of cell movement in any given direction; (3)

the probability distribution of turn angles that will decide the next direction of cell movement; (4) the

frequency of cell stops; and (5) the duration of cell stops. The ultimate direction of cell movement should

also be obtained to check for spatial heterogeneities or chemotactic phenomena.

When such a detailed description of the migration process is desired, models based on Markov chain

concepts must be used to analyze the cell trajectory data [146–148]. At any time t, a migrating cell

can exist in either a directional state (if it moves in a certain direction) or the stationary state (if it has

stopped moving). Clearly, there is a change of state every time the migrating cell changes direction and

when it starts or stops moving. The central assumption here is that state changes are random and do

not depend either on the past history of the cell or on the length of time the cell spent in its current

state. Under these assumptions, the sequence of states is a stochastic Markov sequence. Details for this

analysis may be found in the monograph of Noble and Levine [147]. One usually allows only a ﬁnite

number of directional states by partitioning the set of all possible directions of movement. Four or

eight directional states are typically considered for two-dimensional walks (in addition to the stationary