# Physics of lumen growth

^{a}Mechanobiology Institute, National University of Singapore, Singapore 117411, Singapore;^{b}Laboratoire Physico Chimie Curie, Institut Curie, Paris Science et Lettres Research University, CNRS UMR168, 75005 Paris, France;^{c}CNRS Biomechanics of Cell Contacts, Singapore 117411, Singapore;^{d}Department of Biological Sciences, National University of Singapore, Singapore 117411, Singapore

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Edited by Satyajit Mayor, National Center for Biological Sciences, Bangalore, India, and approved April 11, 2018 (received for review January 8, 2018)

## Significance

The development of intercellular cavities (lumens) is a ubiquitous mechanism to form complex tissue structures in organisms. The generation of Ciona Notochord, the formation of Zebrafish vasculature, or the formation of bile canaliculi between hepatic cells constitute a few examples. Lumen growth is governed by water intake that usually results from the creation of a salt concentration difference (osmotic gradients) between the inside and the outside of the lumen. During morphogenesis or in diseases, lumens can also leak due to improper maturation of the cell junctions that seal them. In this paper, we theoretically describe different conditions and dynamical regimes of lumen growth based on the balance of osmotic pressure, fluid intake, and paracellular leak.

## Abstract

We model the dynamics of formation of intercellular secretory lumens. Using conservation laws, we quantitatively study the balance between paracellular leaks and the build-up of osmotic pressure in the lumen. Our model predicts a critical pumping threshold to expand stable lumens. Consistently with experimental observations in bile canaliculi, the model also describes a transition between a monotonous and oscillatory regime during luminogenesis as a function of ion and water transport parameters. We finally discuss the possible importance of regulation of paracellular leaks in intercellular tubulogenesis.

Epithelial lumens are ubiquitous in organs. They originate from cavities or tubes surrounded by one (seamless lumen) or multiple cells (1). Ions and other bioactive molecules are secreted into the cavities and, if the lumen is open, flow with the physiological medium. The creation of the lumens originates from several classes of morphogenetic events (1). In the case of closed lumens (such as acini, blastocytes, and canaliculi), ion secretion into the forming cavity creates an osmotic pressure. This results in the passive transport of water into the lumen (most often mediated by aquaporins), which constitutes a major driving component for lumen expansion. This osmotic pressure hypothesis was experimentally proposed in the 1960s (2⇓–4). The expansion is mechanically restrained by periluminal tension. In the case of multicellular lumens [e.g., cysts (5⇓–7)], tension results from the contraction of the cells surrounding the lumen. In the case of the intercellular domain, the tension arises from the cortical actin layer surrounding the cavity (8).

Fig. 1*A* illustrates a lumen separating adjacent membranes between two primary rat hepatocytes (liver cells). The contact area between both cells presents an intercellular cleft of around 30 to 50

However, this process is rather generic for many kinds of lumen such as Ciona Notochord lumen (1, 13, 14) or kidney lumens (15). Fig. 1 *B* and *C* also shows that the steady shape of the lumen depends on the secretory activity, which is boosted by the addition of Ursodeoxycholic acid (UDCA). The growth of the lumen can either be monotonous (Fig. 1*C*) or pulsatile (Fig. 1*D*) depending on the periluminal tension and secretory activity. A steady secretion in a closed lumen implies the concomitant existence of leakage. Its nature is likely paracellular (through the nanometer cleft between cells). In the case of multicellular lumen, a few models and experimental studies have considered the role of leaks [originating either from the rupture of cell–cell contacts (7) or permeation across the endothelial layer (16)] during the growth of the lumen. For intercellular lumens, however, the morphogenetic consequences of the leak modulation by the paracellular cleft property have hardly been investigated, either experimentally or theoretically.

Here, we provide a theoretical quantitative study on the balance between secretory activity, leak, and mechanics that determines canaliculi nucleation and growth. Our minimalistic description of lumen expansion identifies the physiologically relevant range of parameters required to establish a stable intracellular cavity and dictate its dynamical properties.

## Modeling Assumption

We consider the lumen as two symmetrical contractile spherical caps (Fig. 2) with a radius of curvature R and a contact angle θ at the lumen edge. The lumen elongates parallel to the cell–cell contact over a distance *SI Appendix*, Table S1) as well as in “international units” based on the estimations derived in *SI Appendix*, *SI(2)*. We study the lumen growth dynamics resulting from the balance between (*i*) the active and passive ion transport across membranes both in the lumen and in the cleft, (*ii*) the passive transport of water along transmembrane osmotic and hydrostatic gradients, (*iii*) the paracellular leakage originating from osmotic gradients and hydrostatic gradients along the cleft, and (*iv*) the mechanical balance controlled by actomyosin contractility. For the sake of simplicity, we considered only one type of anion/cation pair with identical transport properties. These simplified assumptions lead us to consider only ion, water, and momentum conservations (i.e., force balance).

### Mechanical Balance.

In the lumen the hydrostatic pressure

In the cleft, Laplace’s law must be modified to account for membrane adhesion [mediated by Cadherin, for example (18)]:*SI Appendix*, *SI(2)*, we estimate that a few tens of a nanometer away from the interfacial region, between the lumen and the cleft, Eq. **2** results in a homogeneous cleft thickness that hardly deviates from **2** will be replaced by a homogeneous cleft thickness e. In the first-order approximation,

The force balance at the intersection of the lumen with the cleft is the generalized Young–Dupré equation:**1** and **3**.

### Ion Conservation.

In the lumen, ion transport occurs by transmembrane fluxes as well as by leakage at the lumen edges.

The number of ions flowing through the membrane per unit of time and unit of area has two distinct origins. First, an “active” flux per unit area

Ions are also passively transported across transmembrane channels. In this case, the flux is proportional to the chemical potential difference. It reads

The conservation of the total number of ions, *N*, in the lumen then reads

In the cleft, the ion density equilibrates within less than a few microseconds across the cleft thickness e (on the order of a few tens of nanometers). Hence, only the ion flux component along the cleft should be considered. The difference in ion concentration in the lumen, compared with the external medium, generates a diffusive flux *D* is the diffusion coefficient of ions. We neglect all convective contributions to the flux based on the small dimensions of the cleft. Under these assumptions and after integration over the constant thickness e, the local- and time-dependent conservation of ions inside the cleft reads*SI Appendix*, *SI(2)*, we show that that the term **4** is the solution of Eq. **5** at

### Volume Conservation.

In view of the absence of an active biological transport of water, the change in volume results solely from passive fluxes. Due to water incompressibility, the rate of volume change is proportional to the flux of water. The passive contribution from transmembrane water permeation is proportional to the water chemical potential difference and reads

In the cleft, the rapid equilibration of the hydrostatic pressure across the cleft justifies the lubrication approximation to estimate the hydrodynamic contribution of volume change by *SI Appendix*, *SI(2)*].

### Strategy to Solve the Equations.

The complete set of equations that we solve is provided in *SI Appendix*, *SI(4)*. To solve the equations, we assume that the parameters of the cytosol and of the external media are constant and homogeneous. We also assume that the variation in ion concentration

Separating the time scales between lumen dynamics (minutes to hours) and the equilibrium of fluxes in the cleft (subseconds) simplifies the problem. Cleft Eqs. **3**, **5**, and **7** are solved in the quasistatic regime. The ion density in the cleft readily stems from Eq. **5**. We then use it as a source term in Eq. **7**. The solution of Eq. **7** leads to the value of **4** and **6**. We thus reduce the problem to three coupled equations that we formally solve using Mathematica. *SI Appendix*, Table S1 summarizes the various parameters of the problem, and we give their ranges in adimensional and real values in *SI Appendix*, *SI(1)*.

## Existence of Steady States

At steady state, the dynamical equations above simplify as follows: We name

### Steady State Mechanical Balance.

The Young–Dupré relation takes the simple form

### Steady State Ion Conservation.

Assuming azimuthal symmetry, the ion conservation in the cleft (Eq. **5**) can be linearized at the first order in the polar coordinates as**9** admits a simple although cumbersome solution in terms of modified Bessel functions, which we give in *SI Appendix*, *SI(3)*.

**4**) then simplifies as*SI Appendix*, *SI(3)*. For the sake of simplicity, we assume here that the pump activity in the cleft equals that of the lumen.

### Steady State Volume Conservation.

In the cleft, Eq. **7** can be simplified in a similar way and writes*SI Appendix*, *SI(3)*].

Whenever the cleft length is longer than both screening lengths, it acts as a volume source for the lumen. In the opposite case (i.e.,

In the lumen, Eq. **6** simplifies as**12** [see *SI Appendix*, *SI(3)*] and taking its value for

This rescaling of the equations reveals that the relevant parameters controlling the lumen are

Low enough pumping activity cannot compensate the leaks. Independently of its original volume, the lumen shrinks and disappears. When the pump activity is higher, the solution displays two branches. The lower branch is unstable and theoretically corresponds to the creation of a lumen through the nucleation of a small-sized cavity inside the cleft. The instability of this solution can be checked directly on dynamical equations, but it can also be understood with the following argument.

Steady state lumens described by lower branches are small

The upper branches correspond to stable solutions for larger lumens **10** and **13** show that in this limit, the paracellular fluxes diverge as the lumen size approaches the size of the junction. This nonlinear dependence of the paracellular leak in this limit enables the stability of the state. The sensitivity to the edge distance is thus governed by the screening lengths *B* shows that small screening lengths (curve 1) result in stable lumens spanning practically the whole cell–cell contact for all pumping activities. Conversely, large screening lengths (curve 3) confine lumens to smaller sizes above a critical pumping activity. One could thus speculate that the ability of lumens from adjacent cell pairs to merge is determined by their ability to reach the cell edges and is hence controlled by the leak properties of the paracellular cleft.

## Lumen Dynamics

The balance between different fluxes not only determines the steady states of the lumen but also affects lumen dynamics. Fig. 1 *C* and *D* shows that lumen growth can be either monotonous or pulsatile, depending on pumping efficiency. Our model suggests that changing the balance between leaks and ion secretion can induce a transition between both behaviors. The periodicity of the experimental pulsations is of the order of tens of minutes. Consequently, we assume a quasistatic mechanical equilibrium in the cleft. We solve Eqs. **4** and **2** as described in *SI Appendix*, *SI(5)*. The time-dependent variables of the problem are the radius of curvature *SI Appendix*, *SI(4)*.

To exemplify the type of behavior predicted by the model, we fixed the screening length to *SI Appendix*, *SI(3)*, we derive an analytical solution in the transition regime in the limit for large enough lumens (*SI Appendix*, *SI(3)*]; their expressions intricately involve all model parameters. However, the crossover limits between the different dynamic behaviors is set by the parameter

We then calculated the time variation of the lumen concentration (Fig. 5). In all cases, the concentration of the lumen decreases as the lumen grows. It oscillates in phase opposition with the lumen radius in the oscillatory regime. Note, however, that the total amount of ions

## Discussion

The situation of a cavity with constant ion secretion and a fixed cortical tension is intrinsically unstable. A steady state can only be achieved upon three nonexclusive conditions: size- or time-dependent cortical tension, size- or time-dependent ion secretion, and/or leaks. The two first conditions are likely to involve specific biological feedback. The incidence of leaks is far less intuitive to understand. The model we propose quantitatively explores the effect of paracellular leakage in the case of intercellular lumen formation. We account for the specific dependence of the leak on the dimensions of the paracellular cleft, and we show that, in the case of bicellular lumens, the leak can play a critical role in controlling lumen size, dynamics, and composition. The model provides a good qualitative agreement with the experimental phenotypes of canaliculi.

An important prediction of the model is the existence of screening lengths *A* shows that when the distance of the lumen to the cell edge is larger than the screening length, the luminal ion concentration is of the same order as *B* shows that the osmotic pressure decreases considerably less than the hydrostatic pressure. This results in lumens with a much higher ion concentration than what is needed to balance Laplace pressure, should the lumen be closed. Our simplifying assumptions minimize the specific biological details that have yet to be accounted for to perform a quantitative comparison with experimental data. In particular, tight junctions act as diffusive barriers for different classes of ions across claudin pores (20, 21). For the sake of simplicity, we account for their activity as a steady factor included in the hydrodynamic resistance of the paracellular cleft. As the tight junctions mature, their contribution to the paracellular leak might become dominant over the simple evaluation, which is based on a hydrodynamic process. In particular, ion flux selectivity, which enhanced junction stability and mechanosensitivity of tight junctions, may then play a role in the homeostasis of lumens.

We also show that a time-dependent cortical tension is necessary to create an oscillatory behavior. In our model, the origin of cortical tension reinforcement stems from cortex dynamics. As previously mentioned, mechanosensitive mechanisms might reinforce cortex contractility by increasing the actomyosin activity in a stress-dependent manner. However, as shown in Fig. 5, the hydrostatic pressure decreases as the lumen grows, and it is not clear where the mechanosensing reinforcement of the cortex would come from within the frame of this model. Although lipid trafficking by endo- and exocytosis (1) is important for lumen growth, our model indirectly accounts for it as a nonlimiting factor of the lumen expansion. Assuming a nonlimiting rate supply of lipids by vesicular transport, their contribution to cortical tension and thus lumen morphology is negligible. We also do not account for vesicular export of bile in cholestasis cases corresponding to a liver-specific problem that would reduce the generality of our description. We indeed propose that the leak-dependent growth of lumens can be extended to understand, at the tissue scale, the direction of growth of the cavities. In the case described here, the lumen edge can only asymptotically reach the contact edge due to the divergence of the paracellular leak when

## Footnotes

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^{1}To whom correspondence may be addressed. Email: virgile.viasnoff{at}espci.fr or jacques.prost{at}curie.fr.

Author contributions: V.V. and J.P. designed research; V.V. and J.P. performed research; S.D. performed the numerical and symbolic calculations; K.G. measured Lumen dynamics; S.D., K.G., Y.Z., V.V., and J.P. analyzed data; and V.V. and J.P. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1722154115/-/DCSupplemental.

Published under the PNAS license.

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