Subsurface pressure profiling: a novel mathematical paradigm for computing colony pressures on substrate during fungal infections

Colony expansion is an essential feature of fungal infections. Although mechanisms that regulate hyphal forces on the substrate during expansion have been reported previously, there is a critical need of a methodology that can compute the pressure profiles exerted by fungi on substrates during expansion; this will facilitate the validation of therapeutic efficacy of novel antifungals. Here, we introduce an analytical decoding method based on Biot’s incremental stress model, which was used to map the pressure distribution from an expanding mycelium of a popular plant pathogen, Aspergillus parasiticus. Using our recently developed Quantitative acoustic contrast tomography (Q-ACT) we detected that the mycelial growth on the solid agar created multiple surface and subsurface wrinkles with varying wavelengths across the depth of substrate that were computable with acousto-ultrasonic waves between 50 MHz–175 MHz. We derive here the fundamental correlation between these wrinkle wavelengths and the pressure distribution on the colony subsurface. Using our correlation we show that A. parasiticus can exert pressure as high as 300 KPa on the surface of a standard agar growth medium. The study provides a novel mathematical foundation for quantifying fungal pressures on substrate during hyphal invasions under normal and pathophysiological growth conditions.

where, s 11 , s 22 and s 12 are incremental stresses developed during displacement of an element from one point to another. These incremental stresses were used to develop equation of equilibrium, which by using linear stress-stress equation, eventually converted into a system of differential equations with displacement as dependent variables,.

SI-2: Derivation of equation 1 19
To derive the equations for this work the derivation steps performed as described previously 19 . In Eulerian representation of coordinate system (deformed coordinate system) the equation of equilibrium of a material element (square shape after deformation as illustrated in SI Fig. 2) is written as We first transformed this equation to the undeformed coordinate system.
The following geometrically nonlinear strain displacement relation (also illustrated in SI Fig. 3) was used to address the large deformation condition for the fungal growth media: We substituted the nonlinear strain displacement relation into the governing equation transformed to the undeformed coordinate system and obtained the following differential equations after several mathematical steps (details of the mathematical derivation not shown): These equations upon further simplification rendered the following:

SI Fig. 3: Deformation (Extension and rotation of element edges)
Next we incorporated the concept of initial and incremental stress state. Following to the previous equations we re-wrote the equilibrium equation (SI Fig. 4) in deformed 1, 2 coordinate system as follows. In contrast to linear elastic theory, the stress state here (in deformed 1, 2 coordinate system) was the summation of the initial stress and the incremental stress (as shown in Figure 5) because it is defined as the rotation of the material point that will cause an additional incremental stress in the deformed coordinate system. This rendered the following after several mathematical steps (not shown here):

SI-3. Boundary Condition
We briefly formulate here the equations for boundary conditions (also illustrated in SI Fig. 4). The step-by-step mathematical process shown here will be self-explanatory from a solid mechanics point of view. Application of the boundary condition is shown in SI Fig. 6 SI Fig. 6: Traction free and displacement boundary conditions

SI-4. Rationale for neglecting vertical pressure and shear stress
The zero vertical pressure and zero shear stress are called traction free boundary conditions in solid mechanics. For our calculations we neglected any effects from shear stresses because there is no applied shear traction on the surface of the agar. The rationale for neglecting the fungal biomass (source of vertical pressure) is discussed below.
Fungal biomass accumulated on the agar media acts as a circular patch loading on the surface. The vertical stress distribution zone in the agar media was be determined by Boussinesq isobar and is illustrated in SI Fig.7.
SI Fig. 7. Boussinesq isobar for circular patch loading displaying 10% isobar profile (red dashed line) As will be evident from this figure, even though the area of influence of the vertical stress (as evident from SI Fig. 7) is greatest at the bottom and lowest at the surface, our QACT images in Fig.1c revealed that wrinkles at the surface were closest to edge of the colony. This distance increased with depth of the medium until at the very bottom of the plate, when wrinkles were formed farthest away from the edge (SI Fig. 8). This strongly suggested the negligible influence of the vertical stress on the observed wrinkle formation. Moreover, we also observed that most of the biomass generated by the colony was accumulated close to centerline symmetrically, rendering a conical shape of the colony. We reasoned that this requires longer anchor length symmetrically close to centerline for the stability of the colony; hence vertical stresses are highest near to the centerline and decrease monotonically with distance from colony edge.
Hence it was very less likely that vertical stresses (colony weight) had any significant role in the observed wrinkle formation at the edge.