Elastic-Fluid Model for DNA Damage and Mutation from Nuclear Fluid Segregation Due to Cell Migration

Abstract

When cells migrate through constricting pores, they incur DNA damage and develop genomic variation. Experiments show that this damage is not due to DNA breakage from mechanical stress on chromatin in the deformed nucleus. Here we propose a model for a mechanism by which nuclear deformation can lead to DNA damage. We treat the nucleus as an elastic-fluid system with an elastic component (chromatin) and fluid component that can be squeezed out when the nucleus is deformed. We couple the elastic-fluid model to the kinetics of DNA breakage and repair by assuming that the local volume fraction of the elastic component controls the rate of damage per unit volume due to naturally occurring DNA breaks, whereas the volume fraction of the fluid component controls the rate of repair of DNA breaks per unit volume by repair factors, which are soluble in the fluid. By comparing our results to a number of experiments on controlled migration through pores, we show that squeeze-out of the fluid, and hence of the mobile repair factors, is sufficient to account for the extent of DNA damage and genomic variation observed experimentally. We also use our model for migration through a cylindrical pore to estimate the variation with tissue stiffness of the mutation rate in tumors.

Introduction

As tumors grow and metastasize, their cells deform to invade normal tissue. It has been shown that constricted migration of cells leads to genomic variation (1). Thus, mutations associated with tumor growth and metastasis can not only trigger cell migration (2, 3, 4, 5), but cell migration can also cause genomic variation (1), potentially leading to a snowballing effect—with diversity limited by additional selective pressures. Here we examine theoretically a recently proposed mechanism for DNA damage due to constricted migration (6, 7), and show that our model for the mechanism describes quantitatively the observed genomic variation (1).

When a cell deforms during constricted migration, its nucleus also deforms (7, 8, 9). This deformation is not high enough to physically break both strands of DNA; observations of cells aspirated into a micropipette show that when cleaved chromatin stretches into a constriction, its cleavage sites extend considerably without losing their integrity (10). Thus, constricted migration does not appear to enhance physical breakage of DNA due to nuclear deformation. How, then, can nuclear deformation lead to DNA damage?

Here we focus on the following mechano-biochemical mechanism for DNA damage. In the nucleus, DNA constantly breaks even in the absence of deformation, but these breaks are constantly fixed by DNA repair factors, which establish a steady state of breakage and repair with very little net damage. The nucleus can be viewed as a solid-like, elastic component (chromatin) immersed in a fluid containing the repair factors. At sufficiently high nuclear deformation, fluid can be squeezed out from the elastic component (7), decreasing the concentration of fluid—and consequently the concentration of repair factors—associated with the chromatin. Naturally occurring breaks are thus repaired at a lower rate, so that nuclear deformation can increase DNA damage. Segregation can occur without the fluid necessarily leaving the nucleus but rupture of the nuclear envelope, which can occur during constriction (6, 7, 11, 12), can cause further damage. Rupture prolongs the time it takes for the concentration of fluid (and repair factors) in the nucleus to return to its original value after the cell has finished migrating, leading to more damage.

In this article, we present an elastic-fluid model of the nucleus, coupled to a model for the kinetics of DNA damage, which depends on the rates of breakage and repair. In contrast to a previously proposed model (7), which used an equilibrium approximation that breaks down when the number of unrepaired DNA breaks changes with time, we describe the kinetics via time-dependent rate equations for DNA breakage. We fit the resulting theory to experimental data on cells migrating through pores (1, 7). Our theory leads to quantitative predictions of DNA damage inflicted by migration. We find that experimental results are consistent with a linear relation between the mutation rate and DNA damage rate. Finally, we combine cell-in-pore results with simple assumptions to estimate mutation rates for cells migrating in tissue.

Materials and Methods

Elastic-fluid model of the nucleus

In our model of the nucleus, the chromatin is described as an elastic mesh immersed in fluid; in the field of soft-matter physics, this is known as a two-fluid model. In the undeformed nucleus, the volume fraction occupied by the elastic mesh is ${\varphi }_0$ and the volume fraction of the fluid is 1–${\varphi }_0$. When the nucleus is deformed, the mesh is compressed and fluid is squeezed outward. The individual molecules comprising the elastic mesh and fluid can be considered incompressible, so that the volume of fluid that is displaced outside the mesh is equal to the volume change of the fluid-containing mesh.

Cells can undergo large deformations during constricted migration so we must go beyond linear elastic theory. For simplicity, the undeformed nucleus is modeled as a uniform elastic mesh immersed in fluid that occupies a cylinder of radius R_n and length L (blue-green cylinder in Fig. 1). We consider the steady-state deformation of the nucleus as it migrates inside a pore of radius R_p; the elastic mesh elongates into a cylinder of radius Λ_rR_n and length Λ_zL (blue region in Fig. 1). For a rigid pore, Λ_r = R_p/R_n. The rest of the volume of the nucleus contains the fluid that is squeezed out from the elastic mesh (green regions in Fig. 1). Because the total volume of the individual molecules comprising the elastic mesh is conserved, the volume fraction of the elastic mesh in the blue region is given by $\varphi$ = ${\varphi }_0$/(Λ_r²Λ_z). The deformation of the fluid-containing mesh along the length of the cylinder, Λ_z, is related to the radial deformation by a power law Λ_z = 1/Λ_r^α, for some exponent 0 ≤ α ≤ 2 that is related to the Poisson ratio ν by α = 2ν/(1 – ν) in the linear regime. The volume fraction of the elastic mesh in the blue region is therefore as follows:

$$\varphi ={\varphi }_0/{\mathit{\Lambda }}_r^{2-\alpha }.$$ (1)

Figure 1.

Model of the cell. (Left) The undeformed nucleus is modeled as a cylinder (for convenience) of uniform elastic mesh immersed in fluid. (Right) In the deformed nucleus (blue and green regions) inside a pore, the elastic mesh (blue) is elongated into a cylinder of radius Λ_rL_n and length Λ_zL. Fluid is squeezed out of the mesh (green) to allow the mesh to compress.

Note that in this model, segregation of fluid and elastic material can occur without the fluid having to leave the nucleus. We assume that nuclear envelope rupture simply prolongs the time over which the fluid is segregated from the chromatin.

Damage, breakage, and repair rates

The rate of DNA damage density, or the number of unrepaired breaks per unit time per volume, is the difference between the rate at which DNA breaks occur per unit volume and the rate at which they are repaired per unit volume. The DNA break density rate is as follows:

$$k_b\left(\varphi \right)={\kappa }_d\left(\varphi +\frac{c_d}{c_0}\left(1-\varphi \right)\right),$$ (2)

where κ_d is a rate constant, c_d is the concentration of nuclease, and c₀ is a fitting constant. The first term arises because the rate of naturally occurring breaks depends on the volume fraction $\varphi$ of chromatin in the blue region of Fig. 1; Irianto et al. (10) suggest that this breakage rate is independent of nuclear shape and mechanical stress. We include the second term to account for nuclease-induced breakage, to describe a set of experiments in which breaks are induced by adding nuclease.

We assume that breaks are repaired when repair factors bind to breaks. The concentration of unbound breaks, c_U, and bound breaks, c_B, change in time as breaks occur and repair factors bind and unbind. Note that the concentration of bound repair factors is the same as the concentration of breaks that are bound to repair factors, c_B. The average concentration of repair factors within all of the fluid is c_r; we assume that this is fixed in time. The concentration of free repair factors surrounding the chromatin (blue region of Fig. 1) is then given by c_r(1–$\varphi$) − c_B. Adsorption of repair factors decreases the concentration of unbound breaks and increases the concentration of bound breaks by converting unbound breaks into bound breaks. Similarly, desorption of repair factors decreases the concentration of bound breaks by converting them into repaired DNA. These considerations lead to the following rate equations:

$$\frac{d{c}_U}{dt}=k_b\left(\varphi \right)-k_{\text{ads}}{c}_U\left(t\right)\left[c_r\left(1-\varphi \right)-c_B\left(t\right)\right],$$ (3a)

$$\frac{d{c}_B}{dt}=k_{\text{ads}}{c}_U\left(t\right)\left[c_r\left(1-\varphi \right)-c_B\left(t\right)\right]-k_{\text{des}}{c}_B\left(t\right),$$ (3b)

where k_ads and k_des are the rate constants for adsorption and desorption of repair factors from breaks, respectively.

We calculate the total DNA damage, D, or the number of unbound breaks formed in the nucleus, by integrating the unbound break density over the volume V_chrom containing chromatin (the volume of the blue region in Fig. 1), as follows:

$$D\left(t\right)=\underset{V_{\text{chrom}}}{\int }\text{d}V{c}_U\left(t\right)\approx c_U\left(t\right)V_{\text{chrom}}=c_U\left(t\right)\frac{V_{\text{nucl}}{\varphi }_0}{\varphi },$$ (4)

where V_nucl is the volume of the nucleus, and ${\varphi }_0$ is the undeformed volume fraction of chromatin within the nucleus.

Results

Comparison to experimental data on migration through rigid pores: summary of experiments

DNA damage and genomic variation was measured in U2OS cells in several experiments that considered the effects of migration through constricting pores, the effect of repair factor knockdown, and the effect of inducing DNA breaks. The U2OS cell line originates from osteosarcoma and this cell line was used in all rigid pore experiments that we compare with here, with the exception of one comparison we make in the Supporting Material with A549 lung tumor cells. In some experiments, the U2OS cells were modified, either by small interfering (si)RNA transfection to knock down one or more repair proteins (i.e., si-{repair factor(s)}), or by adding nuclease that induces DNA breaks. Details of the experiments are given in Irianto et al. (1, 7), but we provide a brief overview here.

Separation of DNA and protein

Cells were aspirated into micropipettes of different sizes to measure how much mobile proteins become separated from the chromatin in constrictions (7). Intensity of DNA versus pore diameter and intensity of mobile proteins versus pore diameter were both measured.

Constricted migration in untreated and repair knockdown cells

In Irianto et al. (1), untreated cells and siRNA cells were placed on the top surface of transwell filters that contain many 3- or 8-μm diameter and 11-μm-length pores; for this article, we performed additional experiments in 5 μm pores. Twenty-four hours later, some cells had not migrated and remained on the top surface; other cells had migrated through the pores and were on the bottom surface. Double-stranded breaks in top cells and bottom cells after 24 h were measured by observing the number of γ-H2AX foci per cell. The phosphorylation of H2AX into γ-H2AX is considered a hallmark of the number of DNA double-strand breaks generated (13, 14). Two types of siRNA cells were studied: si4 cells that target four different repair proteins, and si-control cells that do not target a particular protein but still have some effect on the cell.

The parameters in the model are fitted using data from these experiments. The fitting is described in the Supporting Material. The fitting parameters are summarized in Table 1.

Table 1.

Summary of Parameters Used in Model for Rigid Pore Migration

Where Parameter Is Used	Parameter	Value	Data Used to Fit Parameters
Solid volume fraction (Eq. 1; Elastic-Fluid Model of the Nucleus)	2–α	0.30 ± 0.02	Fig. 2 C of Irianto et al. (7)
	${\varphi }_0$	0.58 ± 0.02
Damage rate, U2OS cells (Eq. 2; Damage, Breakage, and Repair Rates)	κd	9.8 ± 0.1 h−1μm−3	Fig. 3 D of Irianto et al. (1) and Fig. 5 E of Irianto et al. (7)
	cd/c0	0 or 3.3 ± 0.3
Damage rate, A549 cells (Supporting Material, “Fitting the Parameters”)	κd	8.4 ± 0.1 h−1μm−3	Fig. S1 D of Irianto et al. (1)
Repair factor binding, U2OS cells ((3a), (3b); Damage, Breakage, and Repair Rates)	cr	2.21 ± 0.03 × 10−4μm−3	Fig. 3 D of Irianto et al. (1)
	kads	8 ± 2 × 104 h−1μm−3
	kdes	55 ± 1 h−1
Repair factor binding, A549 cells (Supporting Material, “Fitting the Parameters”)	cr	2.38 ± 0.03 × 10−4μm−3	Fig. S1 D of Irianto et al. (1)
	kads	2.2 ± 0.15 × 104 h−1μm−3 Fig. S1 D of Irianto et al. (1)
	kdes	55 ± 1 h−1
Rupture reentry time (for integrating (3a), (3b); Damage, Breakage, and Repair Rates)	τrupture	2.5 ± 1.6 h	Fig. 5 E of Irianto et al. (7)
Relation between mutation rate and DNA damage rate (Eqs. 5 and 6; Relation between Mutation and DNA Damage)	γ	1.78 ± 0.50	Fig. 5, D and F, of Irianto et al. (1)
	kmthres	0.65 ± 0.13 h−1
	kn	8 ± 2 × 10−5 h−1

The value of c_d/c₀ is 0 in experiments where no nuclease is added and 3.3 ± 0.3 in experiments with added nuclease in Irianto et al. (7).

Predictions for repair factor knockdown experiments

We first consider the effect of migration on DNA damage in repair factor knockdown cells, which have a lower concentration of repair factors, c_r, in our model than untreated cells. Our model predicts 1) repair knockdown cells that do not migrate experience more damage than untreated cells that do not migrate, and 2) migration causes less damage to repair knockdown cells than to untreated cells. This seemingly paradoxical result arises because squeeze-out of fluid due to migration has less effect when the repair factor concentration is initially lower.

Fig. 2 shows the model’s prediction of DNA damage after 3 h migration and 2.5 h rupture recovery through different deformed nuclear diameters (note that the deformed nuclear diameter is equal to the pore diameter when the pore diameter is less than the undeformed nuclear diameter) in comparison with experimental observations from Irianto et al. (1) of U2OS cells 24 h after being placed on a 3-μm transwell filter and also 8 μm for untreated cells transwell filter. We also performed experiments on 5-μm transwell filters to compare with the theoretical predictions. Observations on the top surface for all cell treatments and the bottom of untreated 3 μm migration cells were used to fit parameters in Supporting Material; the model’s predictions of damage in all cells on the bottom of 5 μm pores, in si-treated cells on the bottom surface of 3 μm pores, and in untreated cells on the bottom of 8 μm pores show agreement with observations on the bottom surface of transwell filters. Importantly, we have not introduced any additional fitting parameters to describe these data so the agreement of the theory with the upper two points at 3 μm and all of the points at 5 and 8 μm in Fig. 2 provide verification of the model. In addition, the model predicts the damage expected at pore diameters not yet studied experimentally.

Figure 2.

DNA damage after 24 h is predicted by a model (lines as indicated in the legend) compared with measurements of γH2AX foci from Figs. 1 E and 3 D of Irianto et al. (1) (points as indicated in the legend). Note that the three points on the right (top surface) and the lowest point on the left were used to obtain fitting parameters in our model, but the other points agree with our predictions with no further fitting. The nucleus is much stiffer than the rest of the cell, so we equate deformed nuclear diameter with pore diameter when pore diameter is less than undeformed nuclear diameter, and with the undeformed nuclear diameter when pore diameter is greater than undeformed nuclear diameter. Fig. S1 shows DNA damage in untreated A549 cells and shows untreated U2OS cells on a smaller scale. To see this figure in color, go online.

Relation between mutation and DNA damage

Observations of U2OS cells in Irianto et al. (1) show that migration through constricting pores causes genomic variation. The total genomic variation increases with the number of times that cells migrate through the pores, and the genomic variation for 3 μm pores is significantly larger than for 8 μm pores.

Here we consider the relation between the mutation rate, k_mut, and the DNA damage rate, $k_{\text{dam}}\left(t\right)=V_{\text{chrom}}\left(d{c}_U/dt\right)$. Double-stranded breaks are repaired over several hours (1), and misrepair could be a cause of mutation, so we suggest the simplest possible relation between the mutation rate and damage rate—a linear relation—and explore whether such a relation is consistent with observations. We propose a relation of the following form:

$$k_{\text{mut}}\left(t\right)=\gamma \left(k_{\text{dam}}\left(t\right)-k_{\text{mthres}}\right)\text{Θ}\left(k_{\text{dam}}\left(t\right)-k_{\text{mthres}}\right),$$ (5)

where γ is a proportionality constant and k_mthres is a threshold damage rate, below which damage does not cause mutations; we include a threshold in our proposed relation because cells that do not migrate have some double-stranded breaks but do not mutate. Here, Θ(x) is the Heaviside step function. This is consistent with observations that mutations are related to chromatin organization (15) and DNA repair (16).

To compare with experiments, we use observations of change in loss of heterozygosity (ΔLOH) in Irianto et al. (1) as a quantitative measure of one class of mutations. A heterozygous locus has two different alleles; LOH is loss of one of these alleles. There is also change in LOH if a homozygous locus becomes heterozygous. ΔLOH does not account for all somatic mutations but we use it as a proxy here.

Fig. 3 A shows our prediction for mutation after three transwell migration cycles through different pore diameters. We compare with measurements of ΔLOH in 3 and 8 μm pores in Fig. 5 F of Irianto et al. (1) to fit the proportionality constant γ = 1.78 and damage rate threshold k_mthres = 0.65 h⁻¹. This threshold indicates that cells can withstand a small amount of damage without resulting mutations.

Figure 3.

Mutation after migration. (A) Shown here is change in ΔLOH after three migration cycles through various pore diameters. We propose a linear relation between mutation rate and DNA damage rate and use experimental data from 3- and 8-μm pore migration in Irianto et al. (1) to fit the constants γ and k_mthres. (B) ΔLOH after a varying number of migration cycles is shown here. The model uses a linear relation between mutation rate and DNA damage rate above a threshold and assumes a nonviability rate that increases with accumulated mutations. To see this figure in color, go online.

We now compare to results after many migration cycles through 3 μm pores, as shown in Fig. 5 D of Irianto et al. (1). Because the model has a mutation rate, this suggests that the total number of mutations is approximately proportional to the number of migration cycles (assuming that the time taken for cells to move through the pore is the same during each cycle). We compare the model with ΔLOH measurements for control cells (no migration), three migration cycles, and 17 migration cycles. We find that the model agrees with the observed data for control cells and for three migration cycles, but that Eq. 5 combined with (3a), (3b) predicts much higher change in LOH for 17 migration cycles than is observed in Irianto et al. (1). A likely reason for fewer mutations being observed is that cells can only sustain a limited number of mutations before they die or cannot be expanded as clones; once cells reach this limit they cannot undergo more migration cycles and are not accounted for in this experiment.

Fig. 3 B shows the model prediction with a nonviability rate that depends on the number of mutations. The number of mutations in the population, M_pop, increases as follows:

$$\frac{d{M}_{\text{pop}}}{dt}=k_{\text{mut}}\left(t\right)N\left(t\right)-k_{\text{nonvia}}\left(t\right)M_{\text{pop}}\left(t\right).$$ (6)

The first term is the increase in mutations due to the mutation rate, which is linearly related to the damage rate in the model. Cells are reseeded on the top surface between each cycle to maintain the same number of cells during each migration cycle, so we approximate the cell population as constant, N(t) = N₀. The second term arises because cells can become nonviable either because they die or because they cannot be expanded as clones, and act as a selective pressure toward mutations that promote growth or prevent death. Thus, the rate at which mutations are lost from the population is proportional to the rate at which cells become nonviable multiplied by the number of cells, N(t), multiplied by the number of mutations per cell, M_pop(t)/N(t). We assume that the rate at which cells become nonviable is proportional to the number of mutations per cell (without a threshold), i.e., mutations cause nonviability: k_nonvia(t) = k_nM_pop(t)/N(t), where k_n is a rate constant. Fitting the model to data in Fig. 5, D–F, in Irianto et al. (1), we find k_n = 8 ± 10⁻⁵ h⁻¹. The model shows a good fit to the experimental data, providing support for our assumptions that the mutation rate increases with damage rate and the cell nonviability rate increases with the accumulation of mutations. Due to nonviability, the model predicts that the change in LOH saturates at high cycle numbers.

Summary of parameters used in model

Table 1 provides a summary of all the fitted parameters used in this model. The sensitivity of the fit on each of these parameters is indicated by the error associated with each value that we report in this summary table. The fitting analysis is described in the Supporting Material.

We have approximated the adsorption rate of repair factors, k_ads, as a constant. This parameter depends on the diffusion of repair factors, and we note that at very high solid volume fractions, crowding will reduce diffusion. This suggests that at very high solid volume fractions the model underestimates the net damage rate. In the 3-μm rigid pore experiments that we compare with, the fluid volume fraction is ∼17%, so we do not approach the limit of all the fluid being squeezed out. Complete squeeze-out is predicted for pores with diameters smaller than 1.7 μm.

The value we obtain from fitting for undeformed solid density, ${\varphi }_0$, is in good agreement with experimental measurements of the porosity of euchromatin in Bancaud et al. (17). The fit for rupture recovery time, τ_rupture, can be compared with data in Fig. 1 of Irianto et al. (7), which shows H2B-mCherry moving from the cytoplasm to the nucleus after migration; the time taken for H2B-mCherry to move back to the nucleus shows excellent agreement with the our fitted value for recovery time. We expect repair factors to be able to bind easily to breaks; this is consistent with the large equilibrium constant obtained from the fitted values for k_ads and k_des. The fitted value for repair factor concentration is much smaller than the typical protein concentrations. This discrepancy may arise from two sources. First, we use a single parameter to describe a repair process that involves various proteins that must come together. Second, we are fitting the model to γ-H2AX counts and there could be multiple double-stranded breaks at each focus. Because (3a), (3b) is nonlinear, a more accurate measurement of the number of double-stranded breaks would also affect the other fitting parameters; however, γ-H2AX is considered as the hallmark for measuring DNA damage (18), so provides the best estimate we can obtain for this model. Returning to the remaining fitted variables, we note that misrepair of DNA breaks should lead to mutations, so γ ∼ $\mathcal{O}$ (1) is reasonable. Finally, a nonzero value of k_mthres is expected because cells that do not migrate have small amounts of DNA damage but do not mutate.

Predictions for migration through tissue

The motivation behind the rigid pore migration studies is to help us understand what happens during migration through tissue. Our model shows that deformation of the nucleus causes damage due to separation of repair factors. To apply our approach to tissues, we must understand how the nucleus is deformed during tissue migration. There are many factors that can affect nuclear deformation during migration through tissue, including the typical pore diameter, distribution of pore diameters, and pore lengths; these are related to the density of extracellular matrix fibers but can also vary with other factors, e.g., cross linking (19) and conditions during collagen polymerization (20). Here we apply our model at the simplest level, treating migration through tissue as migration through a cylindrical pore of the same diameter as the average pore diameter in tissue. Another factor is the deformability of the pore (tissue elastic modulus). If the nucleus is stiff enough, then we must take the deformability of the pore into account in calculating the nuclear deformation. Elasticity theory can be used to calculate the pore deformation by matching the stress on the pore with the stress on the nucleus if we know the elastic properties of the tissue and the undeformed pore diameter. Here we have assumed that the nucleus fills the pore, squeezing out the rest of the cell, which is a reasonable assumption for cells aspirated into micropipettes (see Fig. 2 of (7)). Again, we approximate the tissue as a cylindrical pore with the same stiffness and average pore diameter.

The radial deformation of a nucleus of Young’s modulus E_n in a cylindrical pore of Young’s modulus E_t with undeformed radius R_p, is ${\mathit{\Lambda }}_r^2\approx \left(\left(2+{\alpha }_n\right)/\left(2+{\alpha }_t\right)+\left(E_t/E_n\right)\right)/\left(\left(2+{\alpha }_n\right)/\left(2+{\alpha }_{\text{t}}\right)+\left(R_n^2/R_p^2\right)\left(E_t/E_n\right)\right)$, where α is defined above Eq. 1. Note that both the undeformed pore diameter, R_p, in the tissue and the nuclear stiffness, E_n, are related to the tissue stiffness, E_t. An increase of collagen in the extracellular matrix both increases stiffness E_t and decreases the average pore diameter, R_p. Swift et al. (21) reports that the relation between collagen levels and tissue stiffness scales as [coll] ∼ E_t^10.9–1.5, with different exponents for different collagen types, for example collagen-1 concentration scales as [coll] ∼ E_t^1.5. From Fraley et al. (19) and Sapudom et al. (22), we estimate that the pore diameter scales with collagen levels as R_p ∼ 1/[coll]^10.9–1.5, giving the following relation between tissue stiffness and pore diameter:

$$R_p=R_0{\left(E_{t0}/E_t\right)}^{1.0-2.3}.$$ (7)

To obtain R₀ and E_t0, we use measurements of pore diameter distribution in lung tumor tissue in Harada et al. (8), and the median pore diameter is R₀ = 3 μm in tissue with stiffness E_t0 = 6 kPa.

The nucleus responds to the stiffness of the tissue by regulating its lamin levels (21); lamins are intermediate filament proteins that give nuclei their rigidity, E_n (23). Here, we consider two possible relations between the nuclear and tissue Young’s moduli: 1) E_n = E_t. This relation has been observed for cells on 2D soft substrates (24). 2) The nuclear stiffness scales as a power law with tissue stiffness with an exponent derived from results presented in (21). Fig. 1 of Swift et al. (21) reports a power law between the lamin A:B ratio and tissue stiffness of lamin A:B ∼ E_t^0.6. Fig. 7 of Swift et al. (21) shows data for elastic modulus of the nucleus versus the lamin A:B ratio, which can be described by E_n ∼ lamA:B^0.34. This suggests the following relation between nuclear and tissue elastic moduli:

$$E_{\text{n}}=E_{\text{n}0}{\left(E_{\text{t}}/E_{\text{t}0}\right)}^{0.2}.$$ (8)

From fitting to data in Swift et al. (21), we find E_n0 = 0.18 for the choice E_t0 = 6 kPa. We find E_n < E_t over the range of tissue stiffness studied, so the nucleus is deformed more using Eq. 8 than if E_n = E_t. There is therefore more DNA damage using Eq. 2 compared to Eq. 1.

In Fig. 4, we use the relation between damage rate and mutation rate obtained in Relation between Mutation and DNA Damage to predict the average number of mutations that cells will acquire for a range of segregation times (migration + rupture recovery time), showing the prediction using nuclear stiffness given by Eq. 8, because this relation was obtained from cells in 3D tissue. The range chosen is arbitrary because we cannot directly compare ΔLOH data to the rate of somatic mutations; the change in loss of heterozygosity is only one contribution to the somatic mutations per Mb measured in tissues. The times are chosen simply to demonstrate that the mutation rate increases with time spent on constricted migration.

Figure 4.

Given here is a model prediction for ΔLOH mutations in cells migrating through tissue for different segregation times (left axis). Nuclear stiffness increases with tissue stiffness according to Eq. 8 and pore radius varies with tissue stiffness as Eq. 7, with a power law exponent 1.65. Segregation times shown are 0 h (pink, solid), 1 h (purple, dashed), 2 h (red, solid), 3 h (black, dotted), and 4 h (purple, solid). Points shown are observations of somatic mutations (right axis) in different tumors, collected from the literature; the blue adjusted points do not include mutations that are accounted for by carcinogen exposure (see Supporting Material). Because somatic mutations/Mb and LOH are different measures of mutations, these trends should be compared qualitatively, not quantitatively. The gray region is where average pore size is larger than the nucleus, so migration does not contribute to mutations in the model.

For tissues with stiffnesses <2.9 kPa (shaded in Fig. 4), the pores are sufficiently large that we predict no significant effect of migration on mutation rate. We further note that the constriction time could vary greatly for different tumors, depending on the thickness of tissue that cells migrate through, the pore diameter, the tissue elasticity, and the fiber alignment (19). A measurement of the time the nucleus spends in a deformed state in tissue would help to refine our results. We note that although the tumors measured could have lifetimes on the scale of months or years, individual cells within the tumor have much shorter lifetimes, and presumably spend a small fraction of their lifetimes in a deformed state. Finally, we assume uniform selective pressure across different tumors, which enters through the second term in Eq. 6. This effect is negligible at the timescale considered in Fig. 4, but we note that this is based on comparisons of U2OS cells in rigid pore experiments, and different tissues and cell histories could lead to different selective pressures.

When pore diameter varies with tissue stiffness but nuclear stiffness matches tissue stiffness, there is significant change in the mutation rate with tissue stiffness (see Fig. S3). In the case where pore diameter varies with E_t and nuclear stiffness scales as Eq. 8, the model predicts an even more pronounced increase in mutations with tissue stiffness (Fig. 4). In Fig. 4, we also show experimental data compiled from the literature (25, 26, 27, 28, 29) in Irianto et al. (6), which shows an increase in somatic mutations with tissue stiffness. Some of the somatic mutations are caused by exposure to carcinogens, and we have corrected for this in the observations of lung tumors and melanoma. Finally, note that we have calculated change in loss of heterozygosity (left axis), whereas the experiments measured somatic mutations per Mb (right axis). These quantities cannot be compared directly to each other, so we make no quantitative claims about the agreement between our model predictions and the tumor data. Nevertheless, the figure shows that both our model and the measurements exhibit the same qualitative trend of increasing mutations with increasing tissue stiffness.

Discussion

We have presented a model that considers how nuclear deformation affects the balance of DNA breakage and repair. We compared the model quantitatively to observations of DNA damage and mutations in cells that do not migrate and in cells that migrate through 3- or 8-μm-diameter rigid pores. The breakage rate has a constant term and a term that depends on added nuclease; the repair rate depends on the concentration of repair proteins that can bind to break sites. The damage rate is the change in the number of unbound breaks, which is the difference between the breakage and repair rate. We proposed that the mutation rate is linearly related to the damage rate above a threshold and demonstrated that this is consistent with observations if the nonviability rate is proportional to the number of mutations. This suggests that cells can tolerate a small amount of DNA damage without gaining mutations.

In the absence of added nuclease, migration causes an increase in the damage rate because repair factors are squeezed away from the DNA when the nucleus is constricted; DNA breaks occur constantly, but when the concentration of repair factors gets too low during constricted migration, the remaining repair factors cannot fix the breaks at a fast enough rate. This mechanism accounts for all the observed damage. While DNA damage can occur simply due to segregation of repair factors without rupture and repair factors leaving the nucleus, rupture of the nuclear envelope exacerbates the damage by increasing the time during which repair factors are separated from the chromatin.

Our model is based on a number of experimental findings and assumptions that we list here. 1) The chromatin can be modeled as an elastic mesh; repair proteins and nuclease are assumed to reside in a mobile fluid phase. This is confirmed by comparing our elastic-fluid model of the nucleus with experiments that show squeeze-out of repair factors by measuring relative intensities of DNA and repair factor protein in the fluid-rich and elastic-solid-rich regions of nuclei deformed by aspiration into micropipettes; see Figs. 2 C and 6 B in Irianto et al. (7). 2) There is a DNA breakage rate that does not increase with mechanical stress, shown by Fig. 2 of Irianto et al. (10), and a repair rate that depends on mobile repair factors binding to the breaks. Damage occurs when the repair rate cannot keep up with breakage. This is verified by comparison to DNA damage data in Fig. 2. This is further supported by comparing experiments in which nuclease was added to induce additional DNA damage to an equilibrium approximation of the model shown in Fig. 6 D in Irianto et al. (7). 3) The breakage rate increases linearly with the concentration of nuclease added (see Supporting Material). This is verified by comparison to nuclease experiments in Fig. 6 D of Irianto et al. (7). Linearity is further supported by a calculation of nuclease binding showing that almost all of the nuclease bind to the chromatin, using estimates of binding and unbinding rates from data in Fig. 4 C of Irianto et al. (7). 4) Mutation rate is linearly related to DNA damage rate above a threshold. We propose such a relation and find that it is consistent with data in Fig. 5, D and F, in Irianto et al. (1). It is also consistent with observations in Supek et al. (16) showing that DNA repair affects mutation rate.

Our model yields a number of key predictions that could be tested experimentally to verify fully our model: 1) Figs. 2 and S1 contain predictions for DNA damage due to migration in pores of different diameters. Agreement between these predictions and experimental data for different pore sizes would constitute a strong verification of component two of our model above. 2) Fig. S2 contains predictions for foci burnout due to nuclease induction and migration in pores of various diameters. Agreement between this prediction and experimental data for different pore size would provide further verification for components 2 and 3. 3) Fig. 3 A contains predictions for mutation rate after three cycles of migration through pores of different diameters. Alternatively, Fig. 3 B contains predictions for mutations through 3 μm pores after multiple cycles of migration. Experimental agreement with either of these predictions would verify component 4 of our model.

In addition, our model points to additional experimental measurements that are needed to develop a quantitative understanding of the effect of tissue stiffness on mutation rate. For example, measurements of the typical time spent by a cell in a tumor in a deformed state are needed, as well as consistent quantitative measures of mutation rate at the cell and tissue levels.

When we apply our model to tissue, there are additional approximations and assumptions that we make: 1) We approximate the pores in the tissue as cylindrical, deformable pores with an elastic modulus equal to the tissue elastic modulus and undeformed diameter equal to the average pore diameter in the tissue. 2) We use data from Swift et al. (21) to estimate the relation between nuclear stiffness and tissue stiffness as Eq. 8. 3) We use data from the literature (8, 19, 21, 22) to estimate the relation between pore diameter and tissue stiffness as Eq. 7. 4) We assume uniform selection pressure across different tissues based on the nonviability rate obtained for U2OS cells migrating through rigid pores. We note that different tissue types could put different selective pressures on cells.

Fig. 4 shows that there is significant variation in the mutation rate with the time that a cell spends migrating in tissue. Experimental data for the amount by which nuclei are deformed as a function of time during tissue migration could easily be incorporated into the theory. Also, we have considered only a static version of the elastic-fluid model, and have not taken into account the dynamics of how fluid is squeezed out as the nucleus is deformed. In the case of migration through tissue, when the shape of the cell is constantly changing as the cell migrates, the dynamics may become more important and there may be an interesting interplay between the dynamics of squeeze-out and the kinetics of breakage and repair.

Even in the absence of further experiments, however, we emphasize that the quantitative agreement between our model and the data of Fig. 2 implies that fluid squeeze-out from the deformed nucleus and rupture during migration, and the associated loss of repair factor activity is sufficient to explain the experimentally observed DNA damage and genomic variation in cells that have migrated through cylindrical pores. Another example where squeeze-out should have a significant effect is in metastasizing cells in the blood stream. Blood capillaries are ∼3 μm in diameter, so the nucleus is deformed by the capillary walls. Constriction in the blood stream could be important for understanding genomic differences between primary tumors and metastatic cells (30).

By fitting our model to experimental data for migration through cylindrical pores and using estimates from the literature of the dependence of pore size and nuclear stiffness on tissue stiffness, we have applied our model to tissue with no additional fitting. Tumor sequencing results analyzed by others suggest that tumors in stiff tissues have more mutations than tumors in soft tissues (6, 25, 26, 27, 28, 29). The trend in our predictions is in agreement with the trend in experimental observations of the mutation rate in tumors in tissues of varying stiffness, as shown in Fig. 4. Note that our model only accounts for mutations caused by migration-induced DNA damage, and does not include mutations due to other factors. Also, we do not know the migration and deformation history of the cells studied in the literature (25, 26, 27, 28, 29). Nonetheless, the fact that the trend in our predictions is in reasonable agreement with experiments suggests that DNA damage due to migration through tissue is an effect that could contribute to genomic variation. Our emphasis on loss of repair factors reflects the importance of repair proteins to cancer development: genome-scale analysis of thousands of tumors has uncovered ∼200 that are recurrently mutated across dozens of cancer types. This list is enriched for DNA repair genes, such as BRCA1 (31, 32). Repair inhibition drives cancer.

For many cancers, some fraction of mutations is attributable to carcinogenic exposure: lung cancer and smoking, for example, or skin cancer and ultraviolet radiation. Hence, migration-induced segregation and rupture should not realistically account for the whole somatic mutation rates of these cancers. However, smoking-related mutations constitute only ∼40% of all somatic mutations in lung adenocarcinoma (25); even after adjusting for smoking, these cancers are still expected to have a 10-fold higher mutational load than cancers that arise in soft tissues like brain. Meanwhile, in melanoma, stiff skin exhibits more copy number changes than soft skin, and these copy number changes increase even faster with stiffness than do somatic mutation rates (33). This trend points to a correlation between chromosome-level mutations and stiffness that cannot be entirely explained away by ultraviolet exposure, which accounts for ∼60% of mutations (33) (see Supporting Material).

An overwhelming fraction of research into the causes of mutations and cancer has focused on biochemical factors. Although there is significant research into physical effects such as radiation damage, mechanical effects have received little attention. The model presented here focuses on the interaction between mechanics and biochemistry: mechanical squeeze-out of fluid from the deformed nucleus affects the biochemistry because key repair factors are squeezed out along with the fluid and no longer have access to DNA breaks. Our theory and the careful comparison of the theory to a systematic set of experiments show that this mechano-biochemical mechanism can lead to significant mutation rates.

Author Contributions

R.R.B. and A.J.L. developed the model. C.R.P, J.I., Y.X., and D.E.D. designed and performed experiments. R.R.B. and A.J.L. wrote the manuscript.

Editor: Anatoly Kolomeisky.