We present a method to classify distributions of polarity membrane domains on the surface of cells by a multipole expansion in terms of their spherical power spectrum. Using this spherical power spectrum, we describe the dominant symmetry of such a distribution of membrane proteins in terms of either predominantly vectorial, nematic or higher-order type. We first illustrate the method using distributions on a sphere, and afterwards show how surface distributions on cells of non-spherical shape can be mapped to this case. For the convenience of the reader, a list of mathematical symbols can be found in S1A Appendix.

Let f(x) with x ∈ S 2 represent an area density on the surface of the unit sphere S 2, i.e., a surface pattern. Similar to the two-dimensional Fourier transform for functions defined on a plane, we decompose the density f(x) into a sum of spherical harmonic modes F_l(x), see also Fig 1C f ( x ) ︸ surface density = ∑ l = 0 ∞ F l ( x ) ︸ spherical harmonic modes . (1) Here, the spherical harmonic mode F_l(x) of degree l is a linear superposition of spherical harmonics of degree l, and can be written as F l ( x ) = ∑ m = - l l f l m Y l m ( x ), where Y l m ( x ) denotes the spherical harmonic of degree l and order m (normalized to unity). The expansion coefficients f l m are unique and can be computed as a scalar product on the unit sphere between the original density f(x) and the spherical harmonics Y l m as f l m = ∫ S 2 d 2 x f ( x ) Y l m * ( x ). Here, integration is performed over the unit sphere S 2 (with respect to the standard Euclidean measure), while the star denotes the complex conjugate. This formula for the expansion coefficients f l m follows from the ortho-normality of the spherical harmonics, i.e., ∫ S 2 d 2 x Y l m Y l ′ Y m ′ * = δ l l ′ δ m m ′.

A visual representation of this spherical decomposition is given in Fig 1C: the zeroth mode F₀ is isotropic and encodes the mean of the surface distribution f(x). The first mode F₁(x) can be represented by a vector that points to the spherical average of the surface distribution [19]. The second mode F₂(x) is related to nematic polarity and will be at the focus of this work. In fact, the nematic axes of cell polarity as well as the co-orientational order parameters to be introduced in this manuscript depend only on this second mode F₂(x). The possible existence of higher modes is indicated. The original distribution can be restored by summing up all modes.

In analogy to Fourier analysis of one-dimensional signals, we define the spherical power spectrum of the surface pattern f(x) as the sequence ||F_l(x)||² for l = 0, 1, 2, …, where ||F_l(x)||² denotes the squared L²-norm of each spherical mode F_l(x) (normalized by the surface area of the unit sphere) | | F l | | 2 = 1 4 π ∫ S 2 d 2 x | F l | 2 , l = 0 , 1 , 2 , … . (2)

The spherical power spectrum defined in Eq (2) satisfies a generalized Parseval’s theorem ‖ f‖ 2 = ∑ l ‖ F l ‖ 2 , where ||f||² = (4π)⁻¹∫d² x|f(x)|², i.e., the total power of the density f(x) equals the summed power of its modes. The spherical power spectrum ||F_i||² is a true invariant of the pattern f(x), i.e., it does not depend on the choice of coordinate system (while the coefficients f l m depend on this choice). Some authors denote the spherical power spectrum also by S_ff(l) = ||F_l||².

Fig 1D and 1E show prototypical vectorial and nematic distributions and their respective spherical power spectra. We visualize surface patterns also as Mollweide projections, an equal-area, pseudocylindrical geographic projection [20]. The spherical power spectrum of the cap-like distribution, shown in Fig 1D, has a clear peak at the first mode, corresponding to a predominantly vectorial polarity type of the surface distribution. In contrast, for a ring-like pattern as shown in Fig 1E (and, analogously, for a bipolar pattern with two antipodal caps, see Fig 1E’), all odd modes of the spherical power spectrum, including the first mode, vanish by symmetry. The power spectrum attains its maximum at the second mode, which classifies these distributions as nematic.

Biological cells are not perfectly spherical. We propose a simple method to project distributions on the surface of star-convex shapes onto a sphere (i.e., we require that shape can be represented as distance from a common center, which is taken to be the origin, as a single-valued function of the solid angle). Our method allows analyzing the anisotropy of surface patterns independent of any anisotropy of cell shape. For hepatocytes, the correlation between cell shape and apico-basal cell polarity is weak, for a discussion see [15].

Specifically, we radially project from the star-convex shape to a unit sphere concentric with the shape, retaining the nominal value of the original distribution, see S1C Appendix for additional details. To each cell with surface distribution ρ(x) of (apical) polarity proteins, we associate the projection f(x) of this distribution on the unit sphere S 2. Examples of this projection for an epithelial tubular cell from kidney tissue and a hepatocyte from liver tissue are shown in Fig 1F and 1G, respectively. Apical plasma membrane domains of these cells from kidney and liver tissue, respectively, were identified by staining fixed cells with anti-CD13 (Novus, cat NB100-64843, rat, 1/500) as reported previously [14, 15]; example raw images are shown in Fig S1A in S1B Appendix. The kidney cell exhibits clear vectorial polarity as reflected by a peak of the spherical power spectrum at the first mode. Such vectorial polarity is expected as kidney cells are regarded to belong to the vectorial cell polarity type also present in sheet-like epithelia [4]. The observation from a typical kidney cell from Fig 1F is confirmed for an ensemble of cells (n = 286); note that the relative magnitudes of spherical power modes exhibit a consistent pattern, whereas their absolute magnitude scales with the square of the projected area fraction of apical membrane.

In contrast, for the hepatocyte, we find a dominant second mode, while the first mode is less pronounced. If spherical power spectra are averaged over a population of cells, we still find a dominant first mode for the case of kidney cells, see Fig 1F, and a pronounced second mode that exceeds the first mode in the case of hepatocytes, see Fig 1G. In fact, the first Fourier mode ||F₁||² was larger than the second mode ||F₂||² for 79% of the kidney cells, but only for 25% of the hepatocytes, see also Fig S1D, panel A in S1C Appendix. If we normalize Fourier modes by the squared mean ||F₀||² of the surface distribution, we observe a characteristic unimodular distribution of the normalized first mode ||F_l||²/||F₀||² for kidney cells, and a characteristic unimodular distribution of the normalized second mode ||F₂||²/||F₀||² for hepatocytes, see Fig S1D, panel B in S1C Appendix. Note that the large ensemble of cells analyzed here may contain segmentation errors. By manually examining hundreds of individual cases, we noticed that rare cases of potential segmentation errors seem to correspond to smaller values for the first and second mode of the Fourier spectrum. We thus expect that the ensemble-averaged spherical power spectra presented in Fig 1F and 1G represent a lower bound for the respective first and second mode, i.e., that the true values could be higher. The analysis of spherical power spectra of apical membrane distribution of kidney cells and hepatocytes (Fig 1F and 1G) highlights the structural difference between these two different cell types. While the concept of vectorial polarity well describes the polarity of kidney cells, hepatic cell polarity prompts for a description in terms of nematic cell polarity that focuses on the second Fourier mode. We introduce the nematic tensor A of the spherical distribution f(x), A = 1 2 ∫ S 2 d 2 x f ( x ) ( 3 x ⊗ x - 1 ) , (3) where 1 denotes the identity tensor with components 1 i j = δ i j, i, j = 1, …, 3. The nematic tensor A encodes the same information as the second multipole F₂(x). More generally, there is a formal link between the spherical modes of order l and the reduced Cartesian multipole moments [21], see also S1D Appendix. The nematic tensor A is closely related to a moments-of-inertia tensor, see S1E Appendix. We order the eigenvalues α₁, α₂, α₃ of A such that α₁ ≥ α₃ ≥ α₂ holds and denote the eigenvectors corresponding to α₁, α₂, α₃ by a₁, a₂, a₃ Motivated by Fig 2A and 2B, we will refer to a₂ as the ring axis and a₁ as the bipolar axis. Below, the axis a₂ will represent an example of a first principal axis used to define co-orientational order parameters, while the axis a₁ will represent the second principal axis. The numbering of axes a₂ and a₁ was chosen to be consistent with [15] (there α_i = σ_i, i = 1, 2, 3).

To qualitatively assess putative spatial patterns of nematic cell polarity, we propose a visualization method in terms of equivalent cuboids, see Fig 2A–2C. Mathematically, the cuboid for a cell is uniquely determined by the condition that its traceless moments-of-inertia tensor should equal the traceless moments-of-inertia tensor of the spherical distribution f(x), see S1E Appendix. Briefly, the edges of the cuboid are parallel to the eigenvectors of the nematic tensor A associated to f(x), while the side-lengths of the cuboid depend on the eigenvalues of A.

Fig 2A shows an idealized bipolar distribution and its equivalent cuboid. Here, the longest edge of the cuboid is parallel to the bipolar axis a₁ of the surface distribution, while the two shorter axes have equal length. Similarly, for an idealized ring-like distribution, the shortest edge of the cuboid is parallel to the ring axis a₂ of the surface distribution, while the two longest edges have equal length, see Fig 2B. We colored opposite faces of the cuboids in red, green, and blue, where red corresponds to the bipolar axis a₁, and blue to the ring axis a₂. Fig 2C shows the cuboid representation of a typical hepatocyte (with apical membrane distribution shown in green).

Next, we visualized spatial patterns of hepatic cell polarity within a liver lobule using the image segmentation of murine liver tissue from [15] (see Fig S1B in S1B Appendix for an example of the original raw image data). With the cuboidal representation introduced in Fig 2A–2C, we can draw an equivalent cuboid for each hepatocyte within a tissue section, thereby visualizing the biaxial nematic cell polarity of each hepatocyte with respect to its apical membrane distribution, see Fig 2D. In this figure panel, part of a liver lobule is shown with characteristic landmarks represented by the portal vein (orange) and the central vein (cyan). In top view, most of the polarity cuboids are faced with their blue side up (indicating an approximately parallel alignment of the ring axis a₂ of hepatocyte polarity with the large veins; for the chosen tissue sample these are approximately parallel to the z axis). The visualization in terms of equivalent cuboids highlights the existence of a lobule-wide pattern of spatial order. The tissue-level alignment of nematic cell polarity becomes even more apparent when polarity fields are locally averaged to reduce fluctuations, see Fig 2D. Next, we will use order parameters from the theory of liquid crystals to quantify the observed spatial patterns of aligned cell polarity.

We can quantify orientational order of nematic cell polarity within a tissue in terms of orientational order parameters (OOP), S, P, D, C, see Fig 3. These order parameters were originally developed for the study of biaxial order in liquid crystals [22, 23]. We briefly review their definition and provide illustrative examples to convey their geometric meaning. A short computer program accompanying this work allows to interactively explore prototypical examples.

Before we present the formal definition of S, P, D, C in Eq (7), we want to motivate the different roles played by the four OOP. For an ensemble of uniaxial objects, each characterized by a single principal axis, S quantifies how well the ensemble of these principal axes are aligned to a common mean direction. The mean direction of this nematic alignment defines one of several reference axes introduced below. In the theory of simple liquid crystals, S is the most widely used OOP (and probably the most important one). Like S, the OOP P is also defined already for an ensemble of uniaxial objects (with a single principal axis each). While S characterizes the strength of nematic alignment of the single principal axis in the ensemble, P characterizes anisotropic fluctuations of this principal axis, i.e., P becomes non-zero if the deviations of the principal axes from the mean direction are skewed in a particular direction. The direction of anisotropic fluctuations defines a second reference axis.

The two other OOP, D and C, are only defined for biaxial objects that are characterized by two principal axes (a third principal axis can be deduced from the two). The OOP C characterizes nematic alignment of the second principal axis towards a second reference axis. Finally, D quantifies an ‘unexpected relation’ between the fluctuations of the first and second principal axis, respectively, with respect to their alignment to a first reference axis of mean alignment. For a maximum-entropy distribution of orientations, constrained to specific numerical values of S, P, C, the OOP D would be zero.

In liquid crystals, built up by an ensemble of anisotropic molecules, a non-zero value of S can arise from the interactions of uniaxial molecules (e.g., molecules approximated as rods) with a uniaxial external field. Here, the axis of the external field sets the direction of mean alignment, i.e., the first reference axis. A non-zero value of P, however, additionally requires either a second external field, orthogonal to the first one, or boundary conditions that break rotational symmetry for rotations around the first reference axis. Non-zero values of D and C obviously require objects that are intrinsically biaxial, i.e., that do not possess rotational symmetry around their first principal axis. While a non-zero value of D can result already from interactions of biaxial objects with an external uniaxial field, a non-zero value of C requires either an external field that is biaxial, boundary conditions that break uniaxial symmetry, or, possibly, biaxial inter-molecular interactions.

Generally, the order parameters S and P characterize order in an ensemble of uniaxial objects characterized by just a single nematic axis. In this case, S and P depend only on the second spherical mode of the distribution of axes and can be computed in terms of the coefficients f l m of an expansion of this distribution into spherical harmonics, provided a special choice of coordinate system is made, see S1D Appendix for details.

We now proceed to the formal definition of OOP, which will later be generalized to the case of co-orientational order parameters (COOP) and applied to quantitatively characterize design principles of liver tissue. We consider an ensemble of cuboids, each characterized by a tripod of principal axes, which we characterize by orthonormal vectors n, m, l. These unit vectors are only defined up to sign, thus any meaningful physical quantity should be invariant under sign flips n → −n, m → −m, l → −l. Formally, each cuboid is said to possess so-called D_2h-symmetry, i.e., it is invariant under line reflections at its principal axes. We distinguish a first principal axis n, as well as a second principal axis m, and a third principal axis l. It is convenient to introduce, for each cuboid, two traceless tensors Q and B that characterize its tripod of axes [23, 24] Q = 1 2 ( 3 n ⊗ n - 1 ) , B = 3 2 ( l ⊗ l - m ⊗ m ) . (4) Here, 1 is the identity tensor and ⊗ denotes the outer product.

If the principal axes n, m, l of each tripod are given by the normalized eigenvectors of a nematic tensor A (e.g., the nematic tensor of a projected surface distribution f(x) of membrane proteins), we can recover A as linear superposition of the two tensors Q and B, see S1J Appendix. The mathematical advantage of the traceless tensors Q and B is that they conveniently allow to compute ensemble averages, 〈Q〉 and 〈B〉. The eigenvalues of these averaged tensors provide important invariants of orientational order R Q T ⟨ Q ⟩ R Q = ( - 1 2 ( S - P ) 0 0 0 - 1 2 ( S + P ) 0 0 0 S ) , (5) R B T ⟨ B ⟩ R B = ( - 1 2 ( D - 3 C ) 0 0 0 - 1 2 ( D + 3 C ) 0 0 0 D ) . (6) Here, R_Q and R_B are rotation matrices that diagonalize 〈Q〉 and 〈B〉, respectively.

In principle, the rotation matrices R_Q and R_B might be different. However, for important special cases, e.g., ensembles of biaxial molecules interacting with simple external fields, both rotation matrices can be chosen equal, R = R_Q = R_B. In this case, the ensemble-averaged tensors 〈Q〉 and 〈B〉 possess the same eigenvector basis, given by three mutually orthogonal unit vectors w, v, u. The physical meaning of w, v, u is that these vectors define mutually orthogonal symmetry axes, such that the statistics of the ensemble of cuboids is invariant under line reflections at these axes. The ensemble of cuboids is said to possess D_2h-symmetry in this case. In the case of D_2h-symmetry, the common eigenvectors w, v, u of 〈Q〉 and 〈B〉 define a director frame of reference axes of the ensemble. Note R_Q = R_B = [u, v, w] (with column-vectors u, v, w).

Using Eqs (5) and (6), we can rewrite S, P, D, C as averaged direction cosines [23] S = 1 2 ⟨ 3 ( n ( i ) · w ) 2 - 1 ⟩ , P = 3 2 ⟨ ( n ( i ) · u ) 2 - ( n ( i ) · v ) 2 ⟩ , D = 3 2 ⟨ ( l ( i ) · w ) 2 - ( m ( i ) · w ) 2 ⟩ , C = 1 2 ⟨ ( l ( i ) · u ) 2 - ( l ( i ) · v ) 2 + ( m ( i ) · v ) 2 - ( m ( i ) · u ) 2 ⟩ , (7) where 〈⋅〉 averages over an ensemble of nematic axes n⁽ⁱ⁾, m⁽ⁱ⁾, l⁽ⁱ⁾ indexed by i, using a fixed tripod of reference axes w, v, u.

In Eq (7), there is ambiguity regarding the ordering of the various axes, n, m, l, and w, v, u. For our specific case, the mapping from the nematic cell polarity axes a₁, a₂, a₃ to the principal axes n, m, l is only defined up to a constant permutation n = a π ( 2 ) , m = a π ( 1 ) , l = a π ( 3 ) , π ∈ S 3 , (8) where S₃ denotes the group of all permutations of the indices (1, 2, 3). Note that the same permutation π must be used for all tripods of nematic axes a 1 ( i ), a 2 ( i ), a 3 ( i ) of the ensemble. Similarly, if e₁, e₂, e₃ denote the common eigenvectors of 〈Q〉 and 〈B〉 (with some fixed ordering), we distinguish a first reference axis w = e_ρ(2) from a second reference axis v = e_ρ(1), and a third reference axis u = e_ρ(3), where ρ ∈ S₃ denotes a permutation of reference axes. (The axes w, v, u are also called director axes [23]).

We distinguish two different choices for π and ρ, which give rise to orientational order parameters (OOP) as commonly used in the theory of liquid crystals [23], and co-orientational order parameters (COOP) introduced here.

A common choice, put forward, e.g., by Zannoni et al. [23] in the field of biaxial nematics, is to chose the permutations π and ρ of principal and reference axes such that | S | is maximal, P ≥ 0 , and C ≥ 0 . (9)

The condition Eq (9) specifies an ordering of both principal and reference axes (S1I Appendix discusses how a permutation of axes affects the OOP). The tensor 〈Q〉 and the scalar order parameters S and P quantify alignment of the first principal axis n, whereas the tensor 〈B〉 and the scalar order parameters D and C quantify the alignment of the second and third principal axes, m and l. We will refer to the values of S, P, D, C corresponding to the ordering of nematic axes specified by Eq (9) as order parameters (OOP) without further specification. Note that different normalization conventions for OOP are in use, an overview can be found in [21]. While OOP are always well-defined, they have a crucial disadvantage: OOP may change discontinuously if system parameters are smoothly varied due to abrupt changes of either π or ρ, see S1J Appendix.

We propose an alternative choice, where the ordering of axes is directly determined by the properties of a nematic tensor A. In the case of surface distributions on a sphere considered here, we take n to point in the direction of the ring axis a₂, and m to point in the direction of the bipolar axis a₁.

We consider the general case, where for each nematic tensor A⁽ⁱ⁾ from an ensemble of tensors indexed by i, we additionally have a second nematic tensor E⁽ⁱ⁾ for each i. Below, we discuss two natural cases of such reference tensors. Let ε₁, ε₂, ε₃ be the eigenvalues of one of the E⁽ⁱ⁾, ordered such that ε₁ ≥ ε₃ ≥ ε₂, and e₁, e₂, e₃ be the corresponding (normalized) eigenvectors. We introduce a tripod of reference axes for each index i as w⁽ⁱ⁾ = e₂, v⁽ⁱ⁾ = e₁, u⁽ⁱ⁾ = e₃, i.e., we chose ρ as the identify permutation, ρ = id. We define co-orientational order parameters (COOP) by generalizing Eq (7) to this general case, where the reference axes u⁽ⁱ⁾, v⁽ⁱ⁾, w⁽ⁱ⁾ are derived from a set of reference tensors E⁽ⁱ⁾ co - S = 1 2 ⟨ 3 ( n ( i ) · w ( i ) ) 2 - 1 ⟩ , co - P = 3 2 ⟨ ( n ( i ) · u ( i ) ) 2 - ( n ( i ) · v ( i ) ) 2 ⟩ , co - D = 3 2 ⟨ ( l ( i ) · w ( i ) ) 2 - ( m ( i ) · w ( i ) ) 2 ⟩ , co - C = 1 2 ⟨ ( l ( i ) · u ( i ) ) 2 - ( l ( i ) · v ( i ) ) 2 + ( m ( i ) · v ( i ) ) 2 - ( m ( i ) · u ( i ) ) 2 ⟩ , (10) where 〈⋅〉 averages over the ensemble of pairs of tripods indexed by i. We propose a scheme to compute reference tensors E⁽ⁱ⁾ for the important case, where the tensors A⁽ⁱ⁾ = A(x⁽ⁱ⁾) depend on spatial position x⁽ⁱ⁾. For each position x⁽ⁱ⁾, we define E⁽ⁱ⁾ = 〈A(x⁽ⁱ⁾)〉_loc using a local average with a “punctured” three-dimensional Gaussian kernel centered at x⁽ⁱ⁾ (excluding the tensor A⁽ⁱ⁾ at the central position x⁽ⁱ⁾), see S1F Appendix for details. The definition of reference axes in terms of local averages provides a robust definition of reference frame if the direction of nematic order varies as function of spatial position. Indeed, the visualization of nematic cell polarity in liver tissue indicates a curved director field of nematic cell polarity on the lobule-level, see Fig 2D and 2E.

Below, we additionally consider a variation of this theme, where the tripod of reference axes n, m, l is not given by a local average, but by a second set of biaxial objects (namely the local anisotropy of the sinusoid transport network in the liver).

The most important difference between the traditional definition of the OOP, S, P, D, C, and our definition of COOP, co−S, co−P, co−D, co−C, is that the permutation π of principal axes, and ρ of reference axes is determined by the orientational order of the ensemble itself for OOP, but prescribed by the eigenvalues of a nematic tensor for COOP. This apparently small change in the mathematical definition, i.e., fixing the order of reference axes, renders co-orientational order parameters (COOP) a robust metric that is applicable also in the case of curved director fields, where classical order parameters (OOP) may change discontinuously.

Below, we will apply these biaxial co-orientational order parameters to quantify the alignment of nematic cell polarity of hepatocytes in liver tissue.

We now apply the framework of biaxial order parameters to quantify lobule-level patterns of nematic cell polarity in mouse liver tissue. We first examine the co-orientational order of the apical nematic polarity of hepatocytes with respect to its own local average. As detailed in the preceding section, we compare the nematic polarity axes of each individual hepatocyte (introduced in Fig 2C) with a local reference frame, given by a local average of the tensors A with a punctured Gaussian kernel (illustrated in Fig 2E). This provides reference axes w = 〈a₂〉_loc, v = 〈a₁〉_loc, and u = v × w at each hepatocyte position. (Mathematically, 〈⋅〉_loc is defined using nematic tensors, see S1F Appendix for details). We choose the first principal axis n to point in the direction of the ring axis a₂, and the second principal axis m to point in the direction of the bipolar axis a₁.

This choice uniquely specifies the four co-orientational order parameters, see Fig 3G. As additional illustration, we show the distribution of nematic cell polarity axes relative to its local reference system, see Fig 3H. We find that the ring axis a₂ (cyan dots) is clustered around the first reference axis w = 〈a₂〉_loc. Correspondingly, the scalar order parameter co−S of uniaxial nematic order is larger than zero. Additionally, we find a statistically significant phase biaxiality with co−P > 0, revealing that fluctuations of the ring axis n = a₂ are biased away from the average bipolar axis v = 〈a₁〉_loc. This phase biaxiality is also visible in the distribution plot on the sphere in Fig 3H. The second principal axis m (bipolar axis a₁, golden dots) also exhibits a weak ordering with a bias towards v (averaged bipolar axis 〈a₁〉_loc, red) and away from w, reflected by positive values of co−C and co−D, respectively. Thus, using co-orientational order parameters that compare nematic axes with a local average (omitting the central cell), we can rigorously assess biaxial order even in the presence of curved director fields. Moreover, these COOP allow to characterize phenotypes quantitatively. Fig S1E in S1G Appendix reports COOP for Integrin-β1 knockdown mice [15], demonstrating that COOP of hepatocyte polarity are substantially reduced in comparison to normal conditions. In these knockdown mice, communication between sinusoidal cells and hepatocytes is impaired due to selective siRNA silencing of a transmembrane ECM receptor in hepatocytes.

We can analyze nematic order of cell polarity not only within an ensemble of cells, but also quantify the mutual alignment between cell polarity and auxiliary anisotropic structures such as transport networks. As example, we analyze co-orientational order between apical nematic cell polarity of hepatocytes, and the local anisotropy of the blood-transporting sinusoidal network [15, 26]. Sinusoids are specialized blood vessels forming a network within the liver lobule [10]. Fig 4A shows a central-line representation of the sinusoidal network.

We determine the local anisotropy of the sinusoidal network in the vicinity of each hepatocyte. Specifically, if d_k are unit vectors parallel to straight network segments, x_k their midpoint positions and l_k their respective lengths, we define nematic tensors S at each hepatocyte position x⁽ⁱ⁾ S ( x ( i ) ) = ∑ k w ( x k - x ( i ) ) l k ( d k ⊗ d k - 1 3 1 ) . (11) Here, w(x) is a weighting function normalized as ∑_k w(x_k − x⁽ⁱ⁾)l_k = 1. We choose w(x) as a binary cutoff with fixed radius of 20 μm around the center of each hepatocyte. The geometric meaning of S can be understood as follows: The eigenvector s₁, corresponding to the largest eigenvalue, characterizes the direction of preferred sinusoid orientation and will be referred to as preferred axis. The eigenvector s₂, corresponding to the smallest eigenvalue, defines the normal to a plane in which sinusoids orientations are preferentially distributed, and will be referred to as plane axis in the following. The biaxial anisotropy of the sinusoidal network with a distinguished plane axis is indicative of a local layered order, where s₂ represents the normal vector of a stack of parallel layers, which characterizes this layered order. Approximately, s₂ is parallel to both the centerline of the portal vein, and the centerline of the central vein. Fig 4B shows the spatial distribution of these nematic axes, using cuboids with equivalent moments of inertia. The pattern of network anisotropy is similar to the averaged pattern of apical cell polarity: The preferred sinusoid axis s₁ represented by the red faces of the cuboids is oriented approximately along the mean direction of blood flow between the portal and central vein (and thus almost not visible in Fig 4B), reported already in [15]. The plane axis s₂ represented by the blue faces of the cuboids is preferentially aligned parallel to the large veins, representing a local layered organization of this network. This anisotropy of the sinusoidal network with a preferred nematic axis and a local layered organization defines a local reference frame, against which we can compare hepatocyte cell polarity. Fig 4C shows the four co-orientational order parameters between apical nematic cell polarity and local anisotropy of the sinusoidal network. We find that the ring axis a₂ of apical cell polarity is approximately aligned with the plane axis s₂ of the local sinusoid anisotropy, corresponding to co−S > 0. In an idealized picture, the rings of apical cell membrane surrounding hepatocytes would be parallel to the parallel layers of the sinusoidal network. The COOP co−P > 0 shows that fluctuations of the ring axis a₂ around s₂ are not random, but are biased towards the third reference axis s₃ = s₁ × s₂ of the sinusoidal network, and thus away from the preferred axis s₁ of nematic alignment of this network. In physical terms, this implies that liquid-crystal order in liver tissue is truly biaxial (exhibiting so-called phase-biaxial order). The other co-orientational order parameters co−D and co−C are close to zero, i.e., we do not find a particular ordering of the bipolar cell polarity axis a₁ relative to the anisotropic sinusoidal network.

The COOP provide insight about the mutual arrangement of the two transport networks in liver tissue, the sinusoidal network and the bile canaliculi network. The apical membrane domains of hepatocytes indicate the contact surfaces between hepatocytes to the bile canaliculi network. The co-alignment of the ring axis a₂ of hepatocytes and the plane axis s₂ of the sinusoidal network quantified by co−S > 0 suggests a sandwich architecture of these two intertwined networks with alternating layers enriched in either bile canaliculi or sinusoidal network. The observed anisotropic fluctuations of a₂ quantified by co−P > 0 could reflect an effective mutual repulsion between the sinusoidal and the bile canaliculi network.

The co-orientational order is also visualized as a spherical distribution plot in Fig 4D, highlighting the biaxial co-alignment between hepatocyte polarity and the anisotropy of the sinusoidal network in liver tissue.

We present a minimal interaction model that can quantitatively reproduce the co-alignment between hepatocyte cell polarity and the biaxially anisotropic sinusoidal network. If we account only for the ring axis a₂ of hepatocytes, the leading order term of an effective interaction energy is dictated by symmetry and reads H= - λ α 2 ( a 2 ⊗ a 2 ) : S , (12) where α₂ ≈ −0.19 is the average eigenvalue corresponding to the ring axis a₂ of the tensors A of nematic hepatocyte polarity, see Eq (3), and S is a nematic tensor that characterizes the local anisotropy of the sinusoidal network, see Eq (11) and S1H Appendix. Further, (a₂ ⊗ a₂):S denotes the contraction of the two tensors A′ = a₂ ⊗ a₂ and S, i.e., A ′ : S = ∑ i , j = 1 3 A i j ′ S i j = ∑ i , j = 1 3 a 2 , i a 2 , j S i j, where a_2,i, a_2,j denote the components of the vector a₂. Thus, we treat the hepatocytes as uniaxial objects, whereas we retain the biaxial anisotropy of the sinusoid network. The choice of Eq (12) is motivated by a general Landau theory of liquid crystals, see [23, 24]. We calculate the order parameters of an ensemble of axes according to the Boltzmann distribution following this Hamiltonian, using inverse sampling. The control parameter λ is measured in units of an effective temperature that mimics dynamic processes that reduce spatial order [27]. We emphasize that we do not consider liver tissue to represent a thermodynamic equilibrium, despite our use of Eq (12). Instead, Eq (12) represents a phenomenological model that addresses a competition between dynamic processes that either generate or reduce spatial order, respectively.

Fig 5A displays computed COOP, together with the regions of order parameters found for the experimental data of liver tissue. We find a range of values of the effective interaction strength λ (shaded gray in Fig 5A), where the minimal model adequately accounts for the experimental observed values of the co-orientational order parameters. Thus, the interaction of the ring axis a₂ of hepatocytes with the local anisotropy of the sinusoidal network is sufficient to account for the observed biaxial co-orientation. Intriguingly, alternative models assuming either an interaction between S and the bipolar axis a₁, or the full tensor A, did not reproduce the observed co-orientational order, see S1H Appendix.

This finding suggests the cartoon picture of sinusoid-hepatocyte co-alignment in liver tissue shown in Fig 5B. We propose that the ring axis a₂ of hepatocytes preferentially aligns parallel to the plane axis s₂ of the local sinusoidal network. Fluctuations of a₂ break axial symmetry and are biased away from the preferred axis s₁ of the sinusoidal network.