Snooping helices: The elastic path finding algorithm of growing hyphae

Significance

Proprioception is the integrated sense of self-movement and body position in complex organisms. Here, we describe a mechanical form of proprioception driving directional choice making in tip-growing helical organisms. We show that C. albicans hyphae utilize their built-in helicity as an environment-scanning mechanism to explore their surroundings and find target surfaces for invasion. When confined to surfaces, hyphae continue producing in-plane oscillatory shapes that promote further invasive behavior. C. albicans’ inherent mechanical instabilities regulate the switching of growth direction and their abrupt directional decisions can be understood as elastic bifurcations of squeezed, confined helices.

Abstract

How living organisms utilize physical mechanisms to sense their environments and make informed decisions is an open question at the interface of biology and physics. In filamentous organisms like fungal hyphae, the decisions are taken by their growing tip cells and later imprinted onto the rest of the multicellular filament. Here we report on the growth and pathfinding of hyphae from the opportunistic fungal pathogen Candida albicans, whose ability to cross intestinal epithelial layers is associated to severe systemic infections in humans. It has been sporadically reported that C. albicans’s hyphae display helical growth inside or on top of agar gels, helicity turning in the latter case into two-dimensional oscillatory shapes. We provide an extended description of oscillatory C. albicans hyphal growth modalities, revealed under various physical confinements thanks to the use of dedicated microfluidic devices and quantitative time-lapse imaging-based analysis. These include sudden sliding events accompanied by curvature switching of the tip portion, resulting in a final oscillatory morphology of the entire filament, and stable curved tips moving against vertical microfluidic channel’s walls. These behaviors are unified under the formalism of growing squeezed helices, in which the final hyphal curved shapes result from an elastic energy minimization of a spatially confined helical portion at the tip followed by a continuous solidification front. Ultimately, the combination of our experimental results and theoretical framework provides an insight into the penetration strategy of C. albicans hyphae, which is essential for the virulence of this fungal microorganism.

Helically coiled filaments are ubiquitous in nature. They are observed at different scales, ranging from molecular structures such as intermediate filaments (1), microtubules (2), flagella (3), bacteria (4) up to multicellular structures such as climbing vines and roots (5). The helical growth of the plant root, named circumnutation by Charles Darwin, has recently been identified as crucial for the exploration and penetration of soil-mimicking substrates (6), which gives this generic geometry of life high biological and ecological relevance. This strategy used by plants is also a source of inspiration in the field of robotics, stimulating the development of motion planning algorithms to improve the performance of soft robots (7). Helical growth is a widely shared behavior outside plants in the tree of life. Early cell divisions can occur in a helical arrangement in a variety of animals, leading to the well-known coiling of the shell in gastropods or the twisted body of the roller mutants of the nematode Caenorhabditis elegans (see ref. 8 for a review).

In Fungi, which share turgor pressure-driven growth with plants, oscillatory growth modalities of hyphae were reported sporadically and coined as diversely as coils (9), zigzags (10), kinks (11), sinusoids or helices (12, 13). The three last cited studies were performed on the human opportunistic fungal pathogen Candida albicans, a benign member of the human microbiota that can turn into one of the deadliest human opportunistic fungal pathogens. C. albicans is a polymorphic fungus that replicates either as a budding yeast or as a $\approx$2 $\mathrm{\mu }$m-diameter filament (hypha) composed of mononucleated compartments. Among the reversible C. albicans switching systems [yeast-to-hypha but also white-opaque (14)] occurring in response to environmental conditions (15), the yeast-to-hypha switch is recognized as a key virulence trait of this important human fungal pathogen (16). Interestingly, C. albicans helical hyphae have been reported during active human onychomycosis, indicating that they may play a role in invasion (17). They also have been observed in vitro in agar-invasion assays, and associated with increased resistance to antifungal agents (18) or with increased virulence in mice models (11), suggesting a possible mechanistic link between helicity and pathogenicity.

Here we propose a unified mechanistic description of C. albicans hyphal oscillatory growth modalities observed under various conditions, implying either adhesion or physical confinement which we fine-tune using microfluidic technologies. Both are instrumental in the invasive lifestyle of C. albicans, as its hyphae may adhere and navigate at the surface of epithelial cells (drawing oscillations, see ref. 19) or experience confinement in, e.g., the crypts of the colon intestine (20).

We show that the repertoire of hyphal growth dynamics and morphology is well captured by the extension of the concept of “squeelix” (squeezed helix), previously formulated by us and coworkers (21), to growing C. albicans hyphae under confinement. In the present model, we consider that a hypha is composed of a frozen (resignated) portion growing with time and an elastically reconfigurable (mechanically frustrated) portion close to the tip region, the prestress of which is fueled by a differential strengthening of the cell wall along a helical trajectory. A dynamic frontier sets the boundary conditions for the frustrated part, whose shape at every growth step results from the minimization of the total elastic energy stored in it. Last but not least our approach captures generic behaviors that promote efficient penetration mechanisms, suggesting that the intrinsic helicity of C. albicans hyphae could be the result of a selection process that favors its virulence.

Results

C. albicans Hyphae Display Oscillatory Behaviors Under Physical Confinement.

We first incorporated C. albicans yeasts (SC5314 reference strain) within a mixture of filamentation-inducing medium and different concentrations of agar, from 0.2 to 5$\%$, producing gels of various elastic moduli in the range of 100 to 2,000 kPa (22). We observed hyphae growing in all directions from the yeasts embedded in the bulk of the gel, reaching both the upper (with air) and lower (with PDMS) interfaces. In all cases, we noticed oscillations: hyphae display helical shapes within the gel, meandering morphologies at the agar–air interface, and in-plane sinusoidal-like oscillations in-between the gel and the PDMS hydrophilic surface (Fig. 1A). Although right-handed helices only have been reported in the quasi-2D situation represented by a cellophane sheet on top of agar (18), we observed both right- and left-handed helical hyphae growing in the volume of agar gels (SI Appendix, Fig. S1 and Movie S1). In the interface configurations (i.e., agar–air and PDMS–agar), C. albicans filaments experience confinement. Due to the presence of a thin water layer, confinement of growing hyphae at the interface agar–air is mediated by the normal forces resulting from water surface tension directed toward the agar surface (this confinement can be destroyed by adding water on top of the gel, preventing curved shapes in favor of straight hyphae). In-plane confinement also rules the growth of hyphae at the PDMS–agar interface. In this case where friction can be easily tuned by the rigidity of agar, we observed an increased proportion of oscillating hyphae, characterized by higher oscillation amplitudes, with increased agar concentration (SI Appendix, Fig. S2). Note that hyphae do not oscillate on hydrophilic PDMS or glass covered with liquid medium devoid of agar. Last, transferring hyphae from such substrates, which are not conducive to cell adhesion (and thus from where they can be easily detached), to a cell culture treated Petri dish restores oscillatory growth similar to that observed in confinement (Movie S2).

Fig. 1.

Oscillatory hyphae in various microenvironments. (A) Scheme of hypha locations in yeasts-seeded agar gels: at agar–air interface (green), in agar bulk (red), and at PDMS–agar interface (blue). The outlines of the images illustrating these situations use the same color code. Top image: calcofluor white (CW) staining (red) of a hyphae growing in a 2$\%$ agar gel. Middle image: The surface of the gel is stained with Dextran (green) and the hyphae with CW (red). Bottom image: phase contrast image of hyphae navigating in-between a 5$\%$ agar gel and a hydrophilic PDMS layer. (Scale bars, 10 $\mathrm{\mu }$m.) (B) Scheme and scanning electron microscopy (SEM) of the microfluidic mold. Our device comprises four inlet-outlets, a central channel terminated by a serpentine and opening laterally on two vertical arrays of microchannel. The diagram in the Inset shows the typical appearance of a chip after inoculation with C. albicans yeasts and hyphal induction. The hydrodynamic design of the chip allows the yeasts injected from the top inlet to be diverted toward the pockets at the channel entrances, where they are trapped, and then extend hyphae inside. (Main SEM image scale bar, 500 $\mathrm{\mu }$m.) The zoomed-in SEM image (delimited by the rectangle area in the main image) shows the two-height microchannel design. (Scale bar, 150 $\mathrm{\mu }$m.) (C) Phase contrast images of hyphae in 10 $\mathrm{\mu }$m wide microchannels of different heights. (Scale bar, 10 $\mathrm{\mu }$m.) (D) Distribution of amplitudes, half-wavelengths, and maximum curvatures (whose expression is $2{\pi }^2\frac{a}{{\lambda }^2}$, with $a$ the amplitude and $\lambda$ the oscillation wavelength) characterizing oscillations as a function of channel height for a channel width of 10 $\mathrm{\mu }$m. (n_hyphae, n_oscillations) $=$ (17,176) for h $=$ 1 $\mathrm{\mu }$m, (19,223) for h $=$ 1.5 $\mathrm{\mu }$m and (11,83) for h $=$ 3 $\mathrm{\mu }$m. Hyphae were analyzed from n $=$ 2 (h $=$ 1 $\mathrm{\mu }$m and 3 $\mathrm{\mu }$m) and n $=$ 3 (h $=$ 1.5 $\mathrm{\mu }$m) experiments; P values from Kruskal–Wallis tests are indicated. (E) Scheme of biheight channel cross-section. Confocal imaging: projected (sum slices) confocal view of CW stained hyphae and 3D view of the 6.5 $\mathrm{\mu }$m high channel portion (blue square outlines). (Scale bar, 5 $\mathrm{\mu }$m.).

In view of these preliminary results, we decided to turn to microfluidic devices to fine-tune physical confinement. We devised microfluidic channels of h $=$ 1, 1.5, and 3 $\mathrm{\mu }$m height and of various widths (from $W$ $=$ 1.5 to 40 $\mathrm{\mu }$m). Considering the 2 $\mathrm{\mu }$m mean hypha diameter, the smallest channel heights should promote strong to moderate squeezing, while the highest height would only ensure loose confinement. In a more sophisticated type of device, channels display alternative heights of 1.5 and 6.5 $\mathrm{\mu }$m in order to successively confine and partly deconfine a same hypha (Fig. 1B). Yeasts are seeded in a central channel through a main inlet. A portion of these cells are diverted toward trapping pockets located at the entrance of microchannels where they quickly start to initiate a germinal tube which will elongate into a long hypha (Fig. 1 B and C).

Hyphae observed in phase-contrast imaging oscillate markedly in microfluidic channels independently of their height (see images of Fig. 1C). However, the mean amplitude, wavelength, and maximum curvature of these oscillations allow us to distinguish between the squeezed conditions from the one where hyphae are only loosely maintained in height. For instance, the maximum curvature is statistically different between h $=$ 1 (or h $=$ 1.5 $\mathrm{\mu }$m) and h $=$ 3 $\mathrm{\mu }$m (see graphs of Fig. 1D). We then turned to biheight microchannels combining moderate confinement and significant deconfinement (see the zoom image of Fig. 1B) in order to follow along a same hypha the effect of a change in z-confinement. We observed in phase contrast imaging a striking difference between hyphal morphologies in each channel portion, the deconfined hypha displaying loose meandering (Movie S3 A and B). When resolved in 3D using confocal imaging (see an example in Fig. 1E), the same projected image reveals a truncated helix, changing, in this particular case, its direction of rotation after interacting with the PDMS ceiling of the chip, which may prove too low for the helix to grow unhindered. These results led us to consider in the following mainly fairly squeezed hyphae for an in-depth study of the static and dynamical oscillatory hyphal shapes.

Statistical Properties of Hyphae: Environmental “Snooping” and Long Range Persistence.

To quantify the statistical behavior of oscillating confined hyphae, we borrowed methods from polymer physics of semiflexible filaments, with the most intuitive observable being their tangent–tangent-correlation function, TTCF$(\mathrm{\Delta }s)$, as a function of the arc-length distance $\mathrm{\Delta }s$. For random, spatially uncorrelated Gaussian-distributed curvature variation along the length, one would expect a purely exponential decay of the tangent correlation—as found in thermally fluctuating semiflexible “wormlike chains” (WLC) (23). However, both in 3D agar gels and on 2D surfaces at the PDMS/agar interface the hyphae robustly displayed a characteristic behavior that is quite different from predictions of the WLC model, see Fig. 2.

Fig. 2.

Directional persistence of hyphae in 2% agar gels. (A) Tangent–tangent correlation (TTCF) functions of hyphae center lines at a PDMS–agar interface, two corresponding hyphae are displayed in (C) with their centerlines traced. The blue-colored hypha oscillates slightly, and maintains its global orientation. The cyan one displays clear oscillatory behavior while also maintaining its global orientation. The average correlation function (black) is fitted with an exponential decay (dotted black) of characteristic decay length 433 $\mathrm{\mu }$m. (B) TTCF of hyphae center lines in agar gel volume, two corresponding hyphae are represented in D. See also Movie S4 A and B. The orange hypha displays helical behavior, while the red one displays an oscillatory behavior taking effect in a virtual plane within the 3D gel. The exponential decay fit in 3D shows a characteristic decay length of 97 $\mathrm{\mu }$m. The 2D and 3D datasets each correspond to a single experiment. The 2D and 3D datasets contain 336 and 44 hyphae respectively.

At short distances the TTCF falls off quicker than linear, resulting in a quadratic drop, so that the correlations at this scale are strongly reduced. However, after this initial drop, the TTCF recovers, crossing over to the typical exponential decay, as expected within the WLC model. The 2D and 3D situations differ by both their exponential decay characteristic length and the range of deviation from this exponential decay. Both are coherent with the fact that an additional dimension can only impair directional persistence. The short-distance oscillatory behavior acts as a deterministic scanning process with a built-in typical wavelength and seems to be utilized to efficiently probe, or snoop about, the close-range environment, while allowing for a strong directional persistence (i.e., a continuation of the initial direction over large distances). This combined short and long distance environment exploration can be characterized as “directionally persistent snooping.”

Rapid Rearrangements of Hypha Tip Portions Generate Oscillations.

In the following, we asked ourselves the question of how the oscillations are dynamically formed in microchannels. The most immediate idea is that the newly emerging apex would continuously draw the oscillations as it moves forward, while the rest of the hyphae remains shape invariant. However, dynamic monitoring of hyphal growth under confinement shows that this is not the case. Fig. 3 (see also Movie S5A and S5B) indicates that, unexpectedly, oscillations are formed by successive sliding of a limited portion of the hypha close to the apex, while the rest of the hypha upstream stays still. In other terms, a growing hypha seems to be constituted of two parts: one reconfigurable below the apex, and a second one, whose length increases continuously, showing an invariant shape that can be described as “solidified” or “frozen.” We estimated the length of the reconfigurable part by computing the normal displacement in-between sliding events along the hypha as a function of the distance to the tip (SI Appendix, Fig. S3). An exponential fit of the mean displacement gives a characteristic decay length of 6.7 $\mathrm{\mu }$m with a threshold of measurable displacement around a length of 25 $\mathrm{\mu }$m, which is interestingly close to the typical length of a mononucleated hypha compartment (24). The latter value also reflects the length devoid of vacuoles at the tip region (SI Appendix, Fig. S4).

Fig. 3.

Curvature is built by successive sliding of a hyphal portion located at the apex. (A) Image of an oscillating SC5314 C. albicans hypha within a 1.5 $\mathrm{\mu }$m high, 10 $\mathrm{\mu }$m wide microchannel. (B) Images: Superimposition of the successive morphologies of the same hyphae at consecutive time points (blue lines). Schemes: selection of the three last successive morphologies highlighting the movement of the reconfigurable portion at the apex.

Any model of a growing hypha should now account for this observation of the existence of a reconfigurable and a frozen part in a growing hypha. We propose such a model in the next sections, unifying our observations in both 2D and 3D within a single framework based on the concept of squeezed helices.

The Squeezed Helix (Squeelix) Model of a Growing Hypha.

Consider a helical Euler–Kirchhoff filament of length $L$, characterized by its arc length parameter $s$, Euler angles $\psi (s)$,$\varphi (s)$ and $\theta (s)$ describing the rotation of the material frame of the filament with respect to the laboratory frame axes (25). Flattening the filament onto a plane constrains the third angle to $\theta =\pi /2$ and its elastic energy is given by (1, 21, 26)

$$\begin{array}{cc}\hfill E& {\displaystyle =\frac{1}{2}{\int }_0^L[B{\left({\varphi }^{\prime }-{\omega }_{1}sin\psi \right)}^2+C{\left({\psi }^{\prime }-{\omega }_3\right)}^2}\hfill \\ \hfill & +B{\omega }_1^2{cos}^2\psi ]ds,\hfill \end{array}$$ [1]

where ${\varphi }^{\prime }(s)=:\kappa (s)$ stands for the filament curvature, $\psi (s)$ designates the twist angle and ${\psi }^{\prime }(s)=:\tau (s)$ the twist, see also Fig. 4A. The two material constants $B$ and $C$ in Eq. 1 are the bending and torsional stiffness, respectively, ${\omega }_1$ and ${\omega }_3$ are the preferred curvature and twist of the unconfined three-dimensional helical filament (21). Note that ${\omega }_1$ can be equated to the maximum curvature of the oscillations measured experimentally.

Fig. 4.

The “solidifying squeelix” model of a growing hypha. (A) Illustration of the emergence of the 2D squeelix model by the confinement of a 3D elastic helical filament. Under confinement, the two Euler angles, $\psi (s)$ and $\varphi (s)$ couple with each other (21). (B) Successive shapes of a rearranging squeelix highlighting the occurrence of characteristic, discrete twist injection events (formation of twist-kinks) that correlate with sudden, intermittent, highly cooperative rearrangements characterized by curvature flipping within the tail portion of hyphae (“swiping”). (C) Illustration of the solidification (“resignation”) process. A part at the tip is frustrated under the confinement, and described by the squeelix model, while the part further from the tip solidifies losing all the prestress with respect to the imposed shape in a process termed resignation.

The filament’s ground state resulting from minimizing the energy (Eq. 1) obeys two coupled equations: the pendulum-like equation i) ${\psi }^{″}+\frac{B{\omega }_1^2}{2C}sin(2\psi )=0$ and ii) $\kappa ={\omega }_{1}sin\psi$ indicating that curvature is slaved by the twist angle in contrast to the unconfined three-dimensional case (where both decouple). In general, depending on the material parameters, a rich variety of equilibrium shapes resembling loops, waves, spirals, or circles exist (1, 21, 26). These shapes can be seen as the result of interacting repulsive conformational defects corresponding to curvature inversion points. In terms of twist, such defects originate from a rapid variation of $\psi (s)$ (reminiscent of a kink) and are called twist-kinks (Fig. 4B). A relevant dimensionless parameter is $\gamma =\frac{4{\omega }_1^{2}B}{{\pi }^2{\omega }_3^{2}C}$ which measures the ratio of bending and twisting energy. For $\gamma >1$, the ground state approaches a twist-kink-free circular arc of radius $1/{\omega }_1$. For $\gamma <1,$ the filament can be populated by twist-kinks whose density is limited by their elastic repulsion energy (21).

Growth model.

To model the growth of a C. albicans hypha, we recalled the observation that a filament is effectively separated into two parts: far from the growing tip, a solidified, shape-invariant part, and near to the tip, a dynamically rearranging part. An intuitive way understanding this division is to introduce the concept of stress-free, but curved reference shapes (far from tip) following the initial frustration of the tip portion/section and its subsequent rearrangement to find an energy minimizing, yet prestressed, position inside the plane. Both sections are markedly curved, yet only the tip proximal section is prestressed and has the potential to rearrange. The transition process from the frustrated to stress-relaxed state of the hyphae, we will refer to as resignation and the demarcation line between them the “resignation point.” The tip section progressively resigns with time, hardens, stops rearranging as it ages away from the tip. We simulated the combined growth-rearrangement-solidification process by minimizing the elastic energy of the rearranging part while progressively fixing (freezing) the shape of the filament along the resignation point, that is moving at same velocity as the growing tip. Each time step of the simulation consists of three iterative substeps: 1) A new short section of length $\mathrm{\Delta }s$ in the reconfigurable portion is generated 2) The total elastic energy of this portion is minimized and 3) A section of same length $\mathrm{\Delta }s$ is transferred from the reconfigurable to the solidified portion (Fig. 4C).

Validation and parameter matching.

When scanning the range of parameters of a squeelix (preferred twist, curvature, and the resignation length), we observe four different domains represented in the phase diagrams of Fig. 5 A and B, established for three different lateral confinements provided by the channel (in plane) width $W$. We chose parameters values estimated from experiments to simulate the growth dynamics of hyphae confined in 10, 5, and 2.5 μm wide channels, with experimental validation for the smallest and largest widths (Fig. 5C–F). To estimate ${\omega }_1$, one can use the curvature expression for a free, static squeelix (26):

$$\begin{array}{c}\hfill \kappa (s)={\omega }_{1}sn\left(\frac{s-s_0}{\lambda \sqrt{m}}|m\right)\end{array}$$

Fig. 5.

The squeelix model captures different observed behaviors of the hypha. (A and B) Phase diagrams established for a channel’s width of 10 $\mathrm{\mu }$m (A), 5 $\mathrm{\mu }$m and 2.5 $\mathrm{\mu }$m (B). Each phase diagram was computed using 12,500 growing squeelix. The resulting array was then smoothed and interpolated to obtain a continuous representation. (C–F) Simulation and comparison with experiments for swiping (C), bistable nosing (E) and nosing (F) (see Movie S10A–D for dynamical simulations). These three squeelix modalities are supported by observations, contrarily to looping (D). The shape at each few growth steps is displayed with a color, from yellow (old) to blue (new). The solidified part is represented in black, as well as the walls of the channels, and the hypha width is represented in light blue. For all simulations, the resignation length $L$ and the preferred curvature ${\omega }_1$ are fixed with $L=6.7\mathrm{\mu }$m and ${\omega }_1=0.5\mathrm{\mu }$m⁻¹. The two other parameters are the channel width and the preferred twist $(W,{\omega }_3)$ of the corresponding helical filament, whose values are respectively (10 $\mathrm{\mu }$m, $0.6\mathrm{\mu }$m⁻¹), (10 $\mathrm{\mu }$m, 0.2 $\mathrm{\mu }$m⁻¹), (2.5 $\mathrm{\mu }$m, 0.18 $\mathrm{\mu }$m⁻¹) and (2.5 $\mathrm{\mu }$m, 0.3 $\mathrm{\mu }$m⁻¹) for (C–F). (G) Nosing along walls and exit through a 3 $\mathrm{\mu }$m wide narrow channel.

where $\lambda =\frac{1}{{\omega }_1}\sqrt{\frac{C}{B}}$, $sn(u|m)$ is the Jacobi elliptic function with a modulus $m$ satisfying $\frac{m}{E{(m)}^2}=\frac{4}{{\pi }^2}\frac{B{\omega }_1^2}{C{\omega }_3^2}$ and $E(m)$ the complete elliptic integral of the second kind. The curvature varies between $\pm {\omega }_1$. From Fig. 1D, 1.5 $\mathrm{\mu }$m high channels, we estimated the maximal curvature for different hyphae; this is a good estimate that directly gives ${\omega }_1=$ 0.5 $\mathrm{\mu }$m⁻¹. Similarly, knowing the Jacobi sn function is periodic with a period $4K(m)$ where $K(m)$ is the complete elliptic integral of the first kind, one can compute $m$ and estimate ${\omega }_3=$ 0.6 $\mathrm{\mu }$m⁻¹ from the wavelength measurements in Fig. 1D (see also SI Appendix, Fig. S5 for the estimation of ${\omega }_3$ using the power spectral density method, giving very close result).

Finally, we estimate the resignation length $L$ as the decay length $\tau$ of the sliding displacements from SI Appendix, Fig. S3. In a very satisfactory manner, the model captures the whole repertoire of hyphal growth behaviors observed under confinement. With the estimated parameters, the swiping domain corresponds to hypha oscillations by sliding (Fig. 5C, see also Movie S5A), already shown in Fig. 3 and Movie S6 for other examples. Nosing (a term coined by Hanson et al. (27), describing the maintenance of a curvature at the growing apex) and looping (Fig. 5D–F) are occurring when the bending energy contribution dominates over twisting energy. Note that decreasing the channel width leads to a gradual shift toward large curvatures (small radii) of the looping domain. Experimentally, looping is the least represented behavior, as it requires the hypha to be able to go over another portion of itself, which is intrinsically limited by the z confinement. Nosing (Fig. 5 E and F, see also Movie S7 A and B), naturally emerges within our model. This kinetically trapped dynamic state results from the incentive of the hypha to change direction by generating a new twist kink, while not being able to do so due to the opposing force of the wall. The rearrangements due to the injection of every new twist kink require a certain minimal lateral (in plane) space to occur. The occurrence of nosing thus increases as the width of the channel decreases, with a predominance for pure nosing at an intermediate width of 5 $\mathrm{\mu }$m. At the swiping–nosing boundary, or for very thin channels, the hypha noses against a wall before switching to the other wall of the channel, creating bistable nosing. Such a behavior is also observed in our experiments (Fig. 5F, see also Movie S8) where the hypha slides for tens of microns on a single wall before switching sides.

Relevance of nosing in path finding.

We observed in large channels that nosing persists when the orientation of the walls change abruptly. Here, the regularly spaced pillars, whose function is to avoid the collapse of the microfluidic chamber, play a similar role as a wall in a thinner channel, by regularly confining laterally the nosing hypha. Strikingly, when the wall opens up a local, narrow gap—figuring a potential exit—the hypha engaging in nosing behavior manages to enter the gap and leave the channel (Fig. 5G, see also Movie S9). The maintenance of curvature at the tip near the point of contact with the wall (i.e., a quasi-perpendicular orientation of the tip tangent), allows the filament to find this opening without the need of any shape reconfiguration. This underlines the evolutionary advantage of nosing as a purely mechanical pathfinding algorithm, allowing to scan any continuity defect on a surface and be ready to invade the medium below through this weak point.

Curvature, turgor pressure, and cell wall asymmetry.

To address the mechanism behind prestress and curvature generation we develop a model represented schematically in Fig. 6A, based on the rotational motion of the Spitzenkorper (Spk)—the vesicle supply center bringing new material to the cell wall (28). We make here the following assumptions:

Fig. 6.

Cell wall asymmetry and modeling. (A) Scheme of the model. The wall material is preferentially incorporated in the vicinity of the Spitzenkörper (Spk), rotating in the apex section. With growth, the material deposit traces the trajectory of the Spitzenkörper. Turgor pressure differentially extends the wall depending on the material deposit, inducing hyphae curvature. Confinement perturbs the uniform rotation leading to arcs of opposite curvature (squeelix effect). (B) Main image: projected (summed slices) confocal image of hyphae stained using calcofluor white (chitin, red) inside a two-height 40 $\mathrm{\mu }$m wide channel filled with dextran (green). Inset graph: intensity of the chitin signal as a function of the distance along the cross-section direction marked by the red colored arrow. (Scale bar, 10 $\mathrm{\mu }$m.) Side images: Cuts along the z-direction at different points on the hypha as indicated by the numbered dotted lines. The cuts labeled 3 and 6 give symmetric localization of the maximum chitin intensity, as identified by the white stars. (Scale bar, 1 $\mathrm{\mu }$m.) (C) Curvature-fluorescence intensity ratio between opposite walls, each data point (gray) corresponds to a different section of 19 hyphae (data of Fig. 1D, 1.5 $\mathrm{\mu }$m channel height). The linear fit appears in red, and the model fit, Eq. 2, in green.

1. The deposition of the cell wall is localized in the proximity of the Spk.

2. In nonconfined helical hyphae, the Spk position is slightly off-symmetry axis and rotates azimuthally at a rate ${\omega }_3$ (via a yet unknown molecular mechanism). In squeezed hyphae, the occurrence of twist kinks leads to a deformed helical trajectory of the Spk, materialized in rather sharp transitions between the opposite sides of the hypha.

3. The imbalance of cell wall material deposition leads to a wall thickness variation, like $W(\theta )=W_0+W_{1}cos\left(\theta \right)$ with $W_1<W_0\ll R$ (with R the hyphae radius) as function of the azimuthal angle $\theta$.

4. A positive turgor pressure p is prestretching the wall. The wall responds to pressure by stretching differentially, inversely proportional to the wall thickness. A differential wall stretching gives rise to hyphae curvature $\kappa$.

This model accounts for the intriguing observation of an asymmetry of the cell wall structure, materialized by its chitin component (Fig. 6B). Our observations show indeed an increase in chitin intensity located on the inside of the arc of hypha meanders, switching rapidly from one side wall to another, in qualitative agreement with our model.

A hypha subject to these conditions has a total energy $E_{\mathit{tot}}=E_{\mathit{turg}}(p,{\epsilon }_m)+E_{\mathit{el}}(\kappa ,{\epsilon }_m)$, consisting of a turgor pressure term $E_{\mathit{turg}}=-p\pi R^2{L}_0{\epsilon }_m$ that depends on the mean axial strain ${\epsilon }_m$ and the initial hypha length $L_0$ and an elastic wall-stretching term $E_{\mathit{el}}$ depending on both ${\epsilon }_m$ and the curvature $\kappa$. The presence of the curvature gives rise to an azimuthal dependence of the local axial strain $\epsilon \left(\theta \right)={\epsilon }_m-\kappa Rcos\theta$ and to the wall elastic energy $E_{\mathit{el}}\left(\kappa \right)=\frac{Y}{2}{L}_{0}R\int W\left(\theta \right){\epsilon }^2\left(\kappa ,\theta \right)d\theta$ with $Y$ the wall’s Young modulus.

Introducing the fluorescence intensity ratio

$$\begin{array}{c}\hfill I_r=\frac{I_{\mathit{max}}-I_{\mathit{min}}}{I_{\mathit{max}}+I_{\mathit{min}}}=\frac{W_1}{W_0}\end{array}$$

and minimizing the total energy with respect to both mean axial strain and curvature results in the final relation

$$\begin{array}{c}\hfill \kappa \left(I_r\right)=\frac{p}{2Y{W}_0}\frac{I_r}{1-I_r^2/2}.\end{array}$$ [2]

Interestingly, the curvature is proportional to ratio of the turgor pressure and Young modulus times wall width and is insensitive to the precise origin of mechanical inhomogeneity—as both a thickening of the wall (change of $W_0$) and a chitin density variation (changing $Y$) lead to the same result. In the considered experimental range of $I_r$, the nonlinearity in Eq. 2 has marginal impact (Fig. 6C). We can thus consider a linear fit of slope $s$, numerically very close to 1 $\mathrm{\mu }$m⁻¹. Assuming 230 nm as the cell wall thickness (29), we obtain a $\frac{p}{Y}=2W_0\times s$ ratio of about 0.5. Assuming $P=3$ MPa (30), this ratio is fully consistent with $Y=6.3\pm$3.4 MPa, a value obtained in a previous work from bending stiffness measurements (31).

Discussion

Helical shapes offer increased robustness of penetration into various environments, as witnessed by the roots of rice plants: the wild strain, which forms helical roots, is more effective at infiltrating through a grid buried into a hydrogel than a mutant whose roots grow straight (6). The helical growth penetration, akin the geometric action of a “corkscrew,” has the advantage of firmly anchoring itself into the material and removing the corkscrew is therefore impossible without counterrotating the “cork.” Thus, a helical hypha of C. albicans that has penetrated a tissue subject to dynamic deformation, such as the intestine, is able of continued growth without the risk of being easily pulled out from its footprint. Another great benefit of the helix lies in its mechanical compliance making it resilient to tearing via the large stretching movements of its living environment. Although outside the scope of our present study, the next step toward testing the advantages of helical hyphae would be to expose them to conditions closer to their natural in vivo settings, by increasing the complexity of the current in vitro environment, using, e.g., gut-on-chip technology (32).

In this work, we have investigated the hidden, mechanically encoded algorithm behind the physical decision making process in growing C. albicans hyphae and show that they navigate their complex environments by turning their tip cells into helical sensory springs. Microfluidic tools were instrumental to control both the lateral and out of plane confinement of hyphae, allowing to fine-tune the level of environment-induced prestresses in the reconfigurable portion close to the hyphal tip. Hyphal growth within microchannels provided clear visual access to nosing behavior and highlighted new classes of behavior such as bistable nosing, thus enabling the scanning of almost all morphologies associated with squeezed (as well as loosely confined) helices. Owing to 2D confinement-induced prestresses, the elastic energy stored in the reconfigurable portion of helices leads the tip cells to select the trajectory and ultimately the morphology of the filament as a whole.

We have proposed a model based on a helical spinning movement of the Spitzenkorper (Spk)—the vesicle cluster and supply center located at the hyphal tip. The Spk has been reported as a very dynamic organelle: it displays sustained transverse oscillations around the hyphae center line (33, 34) during their forward growth and its position correlates with changes of growth direction (33). Furthermore, the Spk is capable of markedly shifting away from the center line to accompany the “nose-down” tip-shape induced by thigmotropism (35). A helical Spk trajectory at the origin of asymmetric deposition of the cell wall constituents, as evidenced by the chitin labeling, appears thus as a plausible source of stress imbalance, leading to the formation of hyphal meanders and arcs. We hope that our theoretical approach will stimulate future explorations of Spk dynamics in C. albicans hyphae and, if confirmed, the unveiling of the mechanisms responsible for its rotary movement.

The helical growth algorithm combines two important evolutionary features: i) improved scanning of the environment at the cellular scale, and ii) directional persistence of filament growth on supercellular scales. The first phenomenon allows for the exploration and invasion of new habitats once they are reached, while in absence of any local environment change, the second phenomenon becomes more beneficial, by allowing the hypha to reach more favorable environments at minimal material and energy consumption cost—by taking the shortest path in space. This behavior is particularly reflected in the tangent correlation function of in-plane oscillatory hyphae. At short scales the tangent correlation function rapidly drops due to the “oscillatory snooping” environment exploration via the reconfigurable tip cell. However, the oscillatory snooping algorithm, being deterministically sinusoidal in nature, restores quickly the initial direction, leading to a recovery of directional persistence at larger distances.

One of the intriguing physical manifestations of the squeelix model is the prediction of nosing behavior (27), consisting in maintaining a curved tip during growth, and only appearing when the tip hits a wall and slides along it. This behavior has also been described in ref. 35 in a situation of similar hyphal confinement as ours, and reported as a maintenance of the angle of approach in relation to the obstacle over time. Somehow, the nosing behavior appears to be a perfect strategy for a hypha to identify gaps along a wall, or weaknesses in a barrier in general, and use them as a gateway for invasion. It could be effectively utilized in the confinement situation provided by the 20 to 30 $\mathrm{\mu }$m internal diameter of intestinal crypts (36) or when navigating at the surface of any epithelia to implement paracellular route (37). The use of organ-on-chip technology may enable these phenomena to be explored deeper in the future and could shed new light on the ability of oscillating hyphae to penetrate and cross 3D epithelial structures.

The pure nosing (in contrast to nosing-swiping) behavior is also a hint toward the presence of a simple form of memory in the growing hypha system. Indeed, owing to its binary decision to nose along the chosen wall (rather than the opposite one), once the channel is exited, the hyphae continue to move in a very similar direction, resuming upward or downward oscillation according to the wall (high or low) previously selected during nosing. The directional information has thus been stored over long times and distances in form of elastic tip deformation imprinted from one tip cell generation to the next one (see Movie S11: nosing hyphae remember initial directions after leaving long, narrow channels).

Though at this point, the described phenomenon of (squeezed) helical growth and navigation, seems suggestive, we are yet to understand the biological processes behind picking the parameters of this helix under specific conditions. Estimates of the parameters, ${\omega }_1$ and ${\omega }_3$, that we can infer from fitting hypha shapes, show that they are themselves regulated and adaptable to the environmental conditions, like chamber height and even its lateral width.

An important experimental observation and central ingredient of our physical model is the existence of a moving frontier between an elastic, reconfigurable fixed length portion of filament at the tip, and a solidified growing part behind. Although such an abrupt transition remains a first-order approximation, the fact that it provides a satisfactory description of our observations suggests the existence of a process of cell wall aging downstream the apex. Fungal cell walls are built from cross-linked polysaccharides ($\beta$-glucans and chitin) and are continuously remodeled through an equilibrium between synthesis and hydrolysis (38), mostly at their growing apex (39) but also all along the hyphal wall (31). Although the changes in cell wall composition and architecture during aging have not yet been studied in hyphae, Silva et al. have recently reported in yeast cells of Cryptococcus neoformans and Candida glabrata a significant thickening of the cell wall in aged cells, accompanied with ultrastructural changes (40). We cannot exclude that either cell wall thickening or increase of cross-linking occurs far downstream of the hyphal tip, as suggested in ref. (39). Closer to the tip up to a distance of about 15 $\mathrm{\mu }$m, Chevalier et al. (41) reported in the filamentous fungus Aspergillus nidulans a monotonous decrease of the Young and surface modulus. This observation is in agreement with the ability of hyphae to bend near the apex to dynamically draw oscillatory growth. Interestingly, we observed that the reconfigurable portion is devoid of vacuole, and that the frontier between the frozen and reconfigurable portions approximately coincides with the edge of proximal vacuole-free zone (SI Appendix, Fig. S4). Large vacuoles are associated to penetrative mechanical force in C. albicans through yet unknown mechanisms (42). We might speculate here that they could contribute to an additional cell-wall rigidification via turgor pressure driven overstretching followed by wall cross-linking.

Here we explored the macroscopic consequences of the helical scanning hypothesis and showed that many of the exploration and decision making events, like directional memory, oscillatory environment scanning, and wall tracking, could be traced back to mechanical information processing encoded in the elastic energy minimization of a constrained and confined helix. This could be yet another natural example for the general phenomenon of “autonomous physical intelligence,” explored in manmade soft-robotics (43), where all the system responses to their complex environments are hard wired and coded within the subtle interplay of mechanical system parameters and geometry.

Materials and Methods

Fabrication of PDMS Substrate and Devices.

PDMS thin films (experiments using agar gel): PDMS is spin-coated on a glass slide at 3,000 rpm, 30 s, for an expected thickness of 30 $\mathrm{\mu }$m. The glass slide is then fixed with dental silicon at the bottom of perforated petri dishes, which were used as a culture chamber. The chamber can be exposed to an air plasma for typically a minute if a hydrophilic property is desirable before agar gel casting. Microfluidic devices: for the fabrication of the mold, a first layer of SU-8 2002 (MicroChem) is spin-coated on a silicon wafer to obtain a final thickness of either 1, 1.5, or 3 $\mathrm{\mu }$m. After a soft bake, this layer of negative photoresist is exposed to UV light thanks to direct laser writing lithography (Heidelberg, $\mu$PG101). Postexposure bake and development with PGMEA are performed to obtain the first layer. Then a second layer of SU-8 2035 is spread out on top of the mold in order to get the macrochambers of 30 $\mathrm{\mu }$m height. It is precisely aligned on the first level thanks to reference crosses on both layers. The mold is then exposed to UV light through a chromium mask (MJB4 mask aligner). The heights of the mold were measured with a mechanical profilometer (Dektak).

In an alternative version of this device, this first layer is itself a two-level layer with thicknesses of 1.5 and 6.5 $\mathrm{\mu }$m. The process of the 1.5 $\mathrm{\mu }$m layer fabrication is then repeated to add a second layer of 6.5 $\mathrm{\mu }$m height, an operation facilitated by the ease with which levels can be aligned with the $\mu$PG101.

The mold is first treated with trichloroperfluorooctylsilane (ABCR GmbH) deposited in vapor phase during 15 min in order to prevent adhesion of PDMS to the mold. Then PDMS is mixed with its curing agent (Sylgard 184 silicone elastomer kit) at the ratio 10:1 (w/w), degassed in bulk, poured on the mold, degassed again, and finally cured at 75 ^°C overnight. The cured PDMS is then peeled off from the mold, inlets, and outlets are punched with a 1.5 mm diameter biopsy punch (Kai Industries). After the usual step of surface activation using an oxygen plasma, PDMS chips are bonded on glass Petri dishes (Fluorodish, VWR).

Overall, the first level corresponds to microchannels while a second, thicker level feeds these channels. It consists of an inlet opening onto a central channel extended by a serpentine and leading to the main outlet, and of two thin lateral chambers located on either side of the central channel connecting the microchannel outlet to the lateral outlets. As in ref. 31, the serpentine was designed in order to equilibrate the flow toward the main outlet and the lateral outlet, allowing a few yeasts to be trapped in the pockets located at the entrance of each microchannel. These microchannels display different widths, from 1.5 to 40 $\mathrm{\mu }$m for further hyphal development.

C. albicans Culture.

Experiments were performed using the SC5314 reference strain of C. albicans. Strains are stored at $-$70 ^°C in YPD medium (1% yeast extract, 2% peptone, and 2% glucose in distilled water (w/v)) supplemented with 25% of glycerol (v/v). They are streaked on YPD agar plates (YPD medium supplemented with 1.5% of agar (15 g/L of agar)) and kept for up to 3 wk at 4 ^°C. Prior to each experiment, a preculture is done: a colony from the plate is put in 3 mL of YPD liquid medium and kept under agitation overnight at 30 ^°C. Typically, 50 $\mathrm{\mu }$L of this solution diluted in 1 mL of YPD corresponds to an optical density slightly below 1, corresponding to $\approx$10⁷ cells/mL. The 5 $\mathrm{\mu }$L volume of this preculture (i.e., $\approx$10⁶ cells) is added to 495 $\mathrm{\mu }$L of the filamentation medium composed of 445 $\mathrm{\mu }$L of synthetic complete medium (i.e., SC medium, containing 0.67% yeast nitrogen base without amino-acids, 2% glucose, and 0.2% amino acid mix (w/v)) and 50 $\mathrm{\mu }$L of N-acetyl-D-glucosamine at 10mg/mL. Overall, the dilution of the preculture is made so that about a thousand of yeasts in the filamentation medium are inserted in the seeding chamber. To label the cell wall, we used calcofluor white staining at 1 mg/mL.

C. albicans Seeding in Agar Gels.

First, large quantities of agar are prepared and kept in their gel state at room temperature in closed containers to prevent evaporation. The gel is liquefied by microwaving and supplemented with N-acetyl-D-glucosamine at a final concentration of 1 mg/mL. The mixture is then cooled down to about 37 ^°C to allow adding C. albicans yeasts without compromising their viability, and maintaining the gel in a liquid state. The gel $+$ yeast solution is deposited on a surface consisting in a thin PDMS layer spin-coated on a glass slide at 3,000 rpm for 30 s (for a final theoretical thickness of 30 $\mathrm{\mu }$m), then cured for at least 2 h at 75 ^°C, and finally either treated by oxygen plasma (same protocol as to bond PDMS devices, see above) to make it hydrophilic or used right after curing.

Optical Imaging.

Images and image sequences under bright-field illumination were obtained using an inverted microscope (Leica DMi8, equipped with a Hamamatsu C11440 camera) and Metamorph software, regulated at 37 ^°C and 5$\%$ CO₂ during time-lapse experiments (interval between each acquisition set to 5 min). We used a 40$\times$ dry objective (HC PC Fluotar 0.8NA) for multiposition acquisition. Fluorescent images were obtained using a Leica TCS SP8 confocal microscope (with LasX software) equipped with an oil-immersion 40$\times$ objective (APO 40$\times$/1.30 oil CS2). We performed deconvolution (Imaris software) to produce the images of Fig. 6B.

Oscillation Analysis (Images).

In Fig. 2C the hyphae are observed on a free surface. To trace their shapes, we use the Ridge Detection (44) plugin for Fiji associated with a custom Python script to manually discard crossings between hyphae, and compute the tangent–tangent correlation functions. On the other hand, Fig. 1D hyphae are observed in microfluidic channels. Automatized gray-level image analysis is proving difficult due to the presence of the channel walls. Our approach consisted in subtracting an image of the channel before hypha growth, in order to eliminate the walls. In that aim, the position of the pre- and postexperiment images are precisely aligned using the Template matching plugin onFijitoachieveaperfectoverlapofthewalls.Theimageis then enhancedon Fiji (using background subtraction, dilate, remove outliers, contrast adjustment, image binarization) toobtainaskeleton.Aftermanual segmentation todiscardcrossingbetweenhypha, analyze particule allows to select individual hyphae to be considered for final analysis. After a last step performed on Fiji, consisting at straightening hyphae (i.e., correcting any large-wavelength curvature), the amplitudes and wavelengths of each hypha are measured automatically using a custom-made Matlab program.

Oscillation Analysis (3D Stacks).

A similar approach is used for 3D stacks. The stack is initially enhanced in Fiji using contrast adjustment, Gaussian blur, thresholding, skeletonization and dilation to obtain a skeleton of the three-dimensional structure. Then, the Simple Neurite Tracer plugin (45) is used to semimanually trace filaments. The tangent–tangent correlation functions are derived after interpolating and smoothing the traces in a separate Python script.

Oscillation Analysis (Movies).

To consistently track and add new points of a growing hypha across the frames of a time-lapse, the preprocessing is very similar, mainly consisting in stabilizing the movie using the plugin Image Stabilizer (46) from Fiji to ensure the hypha will be the only mobile object. Then, using the Segment Anything Model 2 (SAM 2) (47), we consistently segment hyphae along movies. We skeletonize each segmented frame using medial axis transform, with eroding and thinning operations. Finally, in a custom python script, we determine the distance of all skeleton points to a fixed point (the base) and to the tip on each frame, for each hypha.

Squeelix Model of a Growing Hypha.

Squeelix model.

The squeelix is modeled as a filament, living in a 2D plane, discretized by $N$ points linked to each other by segments of fixed, constant length $\mathit{ds}$. The starting point is fixed, with a local orientation ${\phi }_0$, and a local preferred curvature direction ${\psi }_0$. Each following point $n$ is described by two varying parameters, the local curvature ${\phi }_n^{\prime }$ and the local twist ${\psi }_n^{\prime }$. Initially, this set of parameters $(\overrightarrow{{\phi }^{\prime }},\overrightarrow{{\psi }^{\prime }})$ is set to arbitrary values, and associated with an energy, a discretized version of (Eq. 1). From there, the goal is to minimize the filament energy with respect to these parameters to find the optimal shape of the filament: the squeelix solution. This is done with the Sequential Least Squares Programming method (from Scipy.optimize.minimize). The choice of such method is justified by three reasons: it converges continuously to the nearest local minimum, making it closer to the fast, but continuous processes that drive the hypha rearrangement, with no large influx of energy to jump to far minimums of energy. It can also handle a few hundred parameters, as well as mathematical constraints on these parameters. Such constraints can be used to model which areas of the environment are accessible to the filaments and which are not, referring to PDMS walls in an experimental context. This is achieved using a scalar map of the plane where positive values indicate accessible areas and negative values indicate inaccessible areas. This allows us to model a large variety of lateral confinements.

Growth.

In order to model the growth process, each time step is divided into two: the growth step and the rearrangement step. During the growth step, a new segment is added to the tip. At the resignation point, another segment hardens, effectively rebasing the frustrated filament on top of it while maintaining the original length. In the subsequent rearrangement step, the frustrated filament adjusts to find its optimal position. Overall, at each time step, the resignated part sets a new boundary condition, which influences the shape of the frustrated part after optimization. This optimized frustrated part then provides a new boundary condition for the resignated part, and the cycle continues iteratively.

Phase detection.

To identify the phase (nosing, bistable, looping, or swiping) of a growing squeelix in a channel, we developed a Python detection method. First, nosing (and bistable) filaments are distinguished by evaluating the fraction of length the filament spends within a close range of a single (or both) walls of the channel. Then, discriminating looping filaments from swiping filaments is done by computing their aspect ratio.

Supplementary Material

Appendix 01 (PDF)

Movie S1.

Confocal 3D stack in 2% agar gel. Stack size: 290 × 290 × 70 μm.

Movie S2.

Timelapse of hyphae collected from glass after a few hours of growth, then transferred to a Petri dish. A 10x HC PL APO 0.4NA dry objective was used.

Movie S3A.

Confocal 3D stacks of hyphae in 10 μm wide bi-height channels.

Movie S3B.

Confocal 3D stacks of hyphae in 40 μm wide bi-height channels.

Movie S4A.

Rotational 3D views of the helical hypha shown in Fig. 2D.

Movie S4B.

Rotational 3D views of the planar hypha shown in Fig. 2D.

Movie S5A.

Timelapse corresponding to Fig. 3A. The channel is 10 μm wide.

Movie S5B.

Timelapse with tracking corresponding to Fig. 3. The channel is 10 μm wide

Movie S6.

Large-field, phase contrast timelapse of hyphae showing sweeping across multiple 10μm wide microchannels.

Movie S7A.

Phase contrast timelapses showing nosing.

Movie S7B.

Phase contrast timelapses showing nosing.

Movie S8A.

Phase contrast timelapse showing bistable nosing.

Movie S8B.

Phase contrast timelapse showing bistable nosing.

Movie S9.

Phase contrast timelapse associated with Fig. 5G.

Movie S10A.

Simulations shown in Fig. 5C.

Movie S10B.

Simulations shown in Fig. 5D.

Movie S10C.

Simulations shown in Fig. 5E.

Movie S10D.

Simulations shown in Fig. 5F.

Movie S11.

Phase contrast timelapse showing the resumption of oscillations after nosing within 1.5 μm wide channels. Hyphae first head up to draw their first oscillation if previously nosing along the upper channel wall, and vice-versa.

Data, Materials, and Software Availability

All supporting code can be accessed at https://zenodo.org/records/19071167 (48), while all other study data are provided within the article and/or its supplementary materials.