Regulation of cell growth is paramount to all living organisms. In plants, algae, and fungi, regulation of expansive growth of cells is required for development and morphogenesis. Expansive growth of plant, algal, and fungal cells (cells with walls) employ the same physical principles. The cell produces active solutes within the protoplast and absorbs water from its surroundings. The influx of water produces turgor pressure that stresses the cell wall. The wall deforms in response to the wall stresses generated by the turgor pressure. The wall deformation is reversible (elastic) for non-growing cells, and both elastic and irreversible for growing cells. Experimental evidence (Green, 1969; Richmond et al., 1980; Bartnicki-Garcia et al., 2000; Dumais et al., 2004; Baskin, 2005) and mathematical models (Gierz and Bartnicki-Garcia, 2001; Dumais et al., 2006; Goriely and Tabor, 2008; Geitmann and Ortega, 2009; Fayant et al., 2010) demonstrate that the shapes of cells with walls are determined by controlling the wall mechanical properties during expansive growth. Complicated growth behavior such as helical growth of the large single-celled sporangiophores of Phycomyces blakesleeanus, and the reversals in helical growth direction that occur during development and bulging, can also be explained by controlling the mechanical properties of the cell wall (Castle, 1942; Roelofsen, 1950; Ortega and Gamow, 1974; Ortega et al., 1974; Yoshida et al., 1980; Goriely and Tabor, 2011).

The sporangiophores of P. blakesleeanus are large, single cells that detect and respond to different environmental stimuli. Some of these stimuli are gravity, ethylene, mechanical stretch, gases, temperature, wind, light intensity, spatially asymmetric distribution of light, and the presence of solid objects (Bergman et al., 1969; Cerda-Olmedo and Lipson, 1987). The sporangiophores respond to all of these environmental and sensory stimuli with symmetric and asymmetric changes in growth rate. Typically, sporangiophores used for experimentation are obtained from vegetative spores (Bergman et al., 1969; Cerda-Olmedo and Lipson, 1987). When inoculated on a suitable growth medium, the spores germinate. Within a couple of days after germination, a mycelium is formed and aerial hyphae (sporangiophores) begin to grow vertically (upward). Sporangiophore development is divided into five stages: I, II, III, IV, and V (Figure 1), where stage IV is further divided into three sub-stages: IVa, IVb, and IVc (Bergman et al., 1969; Cerda-Olmedo and Lipson, 1987).

The stage I sporangiophore appears as a single pointed tube that grows longitudinally (Figure 1). All the growth occurs in a short zone (growth zone) between the apical tip and approximately 1.0 mm below it. In general, the mechanical extensibility of the cell wall in the growth zone is greater than in other regions of the sporangiophore stalk. The cell wall in the growth zone elongations (5–20 μm min⁻¹) and twists in such a way that a small particle attached to the surface near the apical tip will rotate (≈30° h⁻¹) in the clockwise direction when viewed from above, and trace out a left-handed helix in time and space (left-handed helical growth). Typically, the stage I sporangiophore reaches a length of 1–2 cm and a diameter of 0.15–0.20 mm near the mycelium.

In stage II, elongation and rotation cease and spherical growth begins at the apical tip (Figure 1). Within 3 h, a bright yellow spherical sporangium is formed (diameter ≈0.5 mm), after which spherical growth stops. In stage III, there is no visible growth (≈2 h in duration). Presumably, this is a period of active spore formation.

Stage IVa begins with elongation and counter-clockwise rotation in a short growth zone on the stalk located approximately 0.6 mm below the base of the sporangium (Figure 1). This right-handed helical growth continues for approximately 1 h before the rotation rate gradually decreases to zero and clockwise rotation begins. Stage IVb begins with the initiation of clockwise rotation and left-handed helical growth (Figure 1). Within an hour or two, the elongation rate, the rotation rate, and the length of the growth zone increase to somewhat constant and typical values of approximately 45 μm min⁻¹, 12° min⁻¹, and 2.5 mm, respectively, and are maintained for hours afterward. Typically, the elongation and rotation rates are measured at one end of the growth zone, usually at the non-growing sporangium, and represent the sum of all the elongation and rotation that occur throughout the length of the growth zone. Stage IVc begins with another reversal in rotation (Figure 1), to the counter-clockwise direction (right-handed helical growth). Stage IVc sporangiophores are long (>10 cm). Stage V sporangiophores are the final stage (Figure 1). They do not exhibit visible growth.

Most biophysical research is conducted with the stage IVb sporangiophore, which is a large cylindrical single-celled stalk (0.1–0.2 mm in diameter) with a spherical sporangium on top (approximately 0.5 mm in diameter) that contains spores for vegetative reproduction. The stalk elongates vertically, opposite to the gravitational acceleration (negatively geotropic), at a nearly constant rate that ranges between 35 and 60 μm min⁻¹. All the growth occurs in a growth zone that is adjacent to the sporangium and typically 2–3 mm in length. The single-celled sporangiophore stalk can grow to a length greater than 20 cm.

Historically, the underlying mechanisms that produce and regulate elongation growth rate of the sporangiophores of P. blakesleeanus have been studied because the sporangiophores are used as a model system for investigations in sensory transduction (Bergman et al., 1969; Cerda-Olmedo and Lipson, 1987). Many sensory responses of stage IVb sporangiophores of P. blakesleeanus are produced by regulating the magnitude of the elongation growth rate (e.g., light growth response, avoidance growth response, house growth response, and stretch growth response) and regulating differential elongation growth rate (e.g., phototropic response, avoidance response, and geotropic response). Prior research demonstrates that wall mechanical properties are altered to regulate elongation growth rate during growth responses (Ortega et al., 1975, 1988, 1989; Ortega and Gamow, 1976, 1977), and both turgor pressure and wall mechanical properties are altered to regulate elongation growth rate during development (Ortega et al., 1991).

It was shown that the in vivo mechanical extensibility of the wall increases during light and avoidance growth responses (Ortega et al., 1975; Ortega and Gamow, 1976, 1977). However, it could not be determined whether the magnitude of the change in mechanical extensibility could completely account for the measured change in elongation growth rate. The results of those investigations demonstrated the need for a quantitative biophysical model for elongation growth rate in order to understand how it is regulated and to elucidate the relationship between the wall extensibility and turgor pressure during regulation. Subsequently, pressure probe methods were developed to determine the in vivo mechanical behavior of the elongating wall during sensory responses (Ortega et al., 1988) and during different stages of development (Ortega et al., 1989, 1991). In those studies, a modified form of the Augmented Growth Equation (Ortega, 1985; Ortega et al., 1989, 1991) was established to provide a quantitative biophysical model for the elongation growth rate of sporangiophores. Within the Augmented Growth Equation, the magnitude and behavior of inclusive biophysical variables are related to biological processes involved in elongation growth (Ortega, 1985, 2004, 2010). The results demonstrate that the magnitudes of the biophysical variables, m_g (longitudinal irreversible wall extensibility of the growth zone) and P_C (critical turgor pressure, sometimes called the yield threshold, Y), are changed to regulate the elongation growth rate during growth responses (Ortega et al., 1988) and during development (Ortega et al., 1989, 1991). Both m_g and P_C are biomechanical variables that determine the magnitude of irreversible deformation rate of the wall. Also, it was found that the turgor pressure, P, changes in magnitude during development (Ortega et al., 1991) but not during the light and avoidance growth responses (Ortega et al., 1988). In addition, it was determined that the magnitude of m_g is a function of both the relative longitudinal irreversible wall extensibility in the growth zone, φ_g, and the length of the growth zone, L_g, although a formal derivation of the relationship was not provided (Ortega et al., 1991).

Prior research that focused on sensory transduction has produced many mutant strains of P. blakesleeanus that exhibit abnormal phototropic responses. These mutant strains are termed mad mutants (Bergman et al., 1969, 1973; Cerda-Olmedo and Lipson, 1987). Bergman et al. (1973) grouped mad mutant strains into three classes based on their geotropic responses, avoidance responses (sometimes called autochemotropic responses), and mycelial responses to light. Because class-two mutant strains are defective in all tropic responses (tropism is produced by differential elongation rate on opposite sides of the growth zone), Bergman et al. (1973) suggested that the mutation was in the output (i.e., growth regulation), but this was not experimentally verified. Class-two mutants are termed “stiff” mutants because of their diminished or non-existent tropic responses. Ootaki et al. (1974) conducted genetic complementation tests on mutant strains from each of the three classes and identified two genes associated with the tested class-two mutant strains, madD and madE. Subsequent research (Ootaki et al., 1977; Ootaki and Miyazaki, 1993) identified additional genes associated with class-two mutants, some of which are used to obtain insight into the mechanisms and interactions of the sensory transduction pathways (Campuzano et al., 1996; Grolig et al., 2000). Thus for the strains tested, stiff mutant strains are defective in genes madD, E, F, G, and J (Campuzano et al., 1996; Grolig et al., 2000).

In the present study, it is hypothesized that the magnitudes of the biophysical variables that regulate elongation rate are different for stiff mutant strains compared to those of wild type (WT). The biophysical variables, m_g, P, and P_C, for stage IVb sporangiophores from stiff mutant strains are determined and compared to respective variables previously obtained from WT stage IVb sporangiophores (Ortega et al., 1989). Two strains of stiff mutants, C149 madD120 (−) and C216 geo- (−), are studied. It is found that the magnitude of m_g is much smaller for the sporangiophores from these stiff mutant strains compared to the WT. This finding appears to present a paradox because the elongation growth rates of stiff mutants and WT sporangiophores are statistically the same magnitude. However, after determining the magnitudes of P and P_C, it is discovered that the magnitudes of P and the difference (P – P_C) are significantly larger for stiff mutant sporangiophores and compensate for the smaller magnitude of m_g. Other experimental results demonstrate that the smaller magnitude of m_g occurs because of a decrease in the magnitudes of both the length of the growth zone, L_g, and relative longitudinal irreversible wall extensibility in the growth zone, φ_g. A decrease in either L_g or φ_g can account for the diminished tropic responses of the stiff mutant sporangiophores.

Previously, it was concluded that a quantitative biophysical model for elongation growth rate of the sporangiophore was needed in order to understand how it is regulated. Therefore in this study, a modified Augmented Growth Equation is formally derived for cells in which expansive growth occurs in a localized region of the cell wall, i.e., in a “growth zone.” Expansive growth in a growth zone is common in fungal sporangiophores (Bergman et al., 1969; Cerda-Olmedo and Lipson, 1987), fungal hyphae (Heath, 1990), root hairs (Dumais et al., 2004), and pollen tubes (Fayant et al., 2010) of plants. Importantly, the derivation of the modified Augmented Growth Equation explicitly produces a relationship between m_g, φ_g, and L_g, and in addition, accounts for the elastic elongation within the growth zone and within the non-growing wall of the stalk adjacent to the growth zone. Also, the derivation recovers previous forms of the modified Augmented Growth Equation that have been used to interpret and analyze the results of prior investigations.

Prior experimental research measured the in vivo mechanical extensibility of the stage IVb sporangiophore’s wall before and during the light and avoidance growth responses (Ortega et al., 1975; Ortega and Gamow, 1976, 1977). The results of those investigations demonstrated the need for a quantitative biophysical model for elongation growth rate in order to investigate its regulation because it could not be determined whether the measured change in mechanical extensibility could completely account for the measured change in elongation growth rate. Biophysical equations describing the elongation growth rate of stage IVb sporangiophores (Eqs 10–12) are derived from the Augmented Growth Equation for wall deformation (Ortega, 1985). The derivation recovers previous biophysical equations used to analyze the results of pressure probe experiments conducted on internode cells of C. corallina, Eq. 5 (Proseus et al., 1999, 2000) and sporangiophores of P. blakesleeanus, Eq. 12 (Ortega et al., 1989, 1991). Equations 10 and 11 describe the elongation growth rate when the turgor pressure is changing. Equation 12 describes the elongation growth rate when the turgor pressure is constant. These equations (Eqs 10–12) are appropriate for cells with a distinct growth zone, e.g., stage I and stage IV sporangiophores, fungal hyphae, pollen tubes, and root hairs (Figure 3B). It is noted that Eqs 3–5 are appropriate for cells without a distinct growth zone, e.g., plant cells in tissue and internode cells of algae (Figure 3A).

An important feature of this derivation is the demonstration that φ (relative irreversible wall extensibility) within Eq. 3 is not the appropriate biophysical variable needed to describe the irreversible properties of the wall in cells with a distinct growth zone, because these cells can regulate the overall irreversible extensibility by changing the length of the growth zone, L_g, as well as the magnitude of wall’s ability to extend irreversible, φ_g. A more appropriate biophysical variable for cells with a distinct growth zone is m_g, which is the product of L_g and φ_g, i.e., m_g = φ_g L_g. Importantly, m_g can be determined directly with pressure probe experiments (Ortega et al., 1989, 1991).

The results presented in Table 1 demonstrate that the magnitudes of m_g, P_C, and L_g decrease while the magnitudes of P and (P – P_C) increase for stage IVb sporangiophores from C216 and C149 strains (stiff mutants) compared to those of WT. The finding that magnitudes of both m_g and P_C decrease for the stiff mutants appears unusual. Previous research that compares the magnitudes of m_g and P_C for stage I and stage IVb sporangiophores indicates that P_C decreases when m_g increases (Ortega et al., 1991), which is consistent with the expected behavior of a simple viscoelastic material. The finding that m_g and P_C both decrease for the stiff mutants suggests that the composition of the growing cell wall and the molecular mechanisms (chemistry) that regulate the magnitudes of m_g and P_C are more complex than that of a simple viscoelastic material. The finding that m_g and P_C of the internode cells of C. corallina are both smaller in magnitude compared to respective values for stage I sporangiophores (Ortega, 2004) supports this suggestion, because the composition and chemistry that regulate the magnitudes of m_g and P_C of the algal walls are different compared to those of the fungal walls.

Interestingly, the elongation rate (dL/dt) for C216, C149, and WT stage IVb sporangiophores are statistically the same magnitude (Table 1). It can be seen with the use of Eq. 12 and the data in Table 1 that decreases in magnitudes of m_g for the stiff mutant sporangiophores are compensated by increases in magnitudes of (P – P_C) to maintain elongation growth rates, dL/dt, that are of similar magnitude measured for WT sporangiophores.

Normalized values can be used to understand the relationship between relevant biophysical variables. The normalized values for dL/dt, m_g, and (P – P_C) presented in Figure 5 demonstrate that magnitudes of m_g and (P – P_C) of stage IVb sporangiophores from C216 and C149 strains decrease and increase, respectively, to maintain elongation growth rates that are of similar magnitudes as those obtained from WT. The decreases in magnitudes of m_g indicate a diminished capability of the wall to deform irreversibility in the longitudinal direction. The increases in magnitudes of (P – P_C) indicate an increase in the magnitude of effective stress within the wall that produces irreversible deformation. Thus for the two stiff mutant strains studied, the irreversible extension of the wall is significantly reduced and should significantly reduce the elongation rate. However, magnitudes of P and (P – P_C) increase, which increases wall stresses that produces irreversible extension for the two stiff mutant strains, and produce elongation growth rates of similar magnitudes measured for WT sporangiophores.

The normalized magnitudes for m_g, L_g, and φ_g presented in Figure 6 demonstrate that decreases in magnitudes of m_g for stage IVb sporangiophores from C216 and C149 strains are produced by decreasing both L_g and φ_g. Decreases in either L_g or φ_g can diminish the sporangiophore’s ability to produce significant differential elongation rates on opposite sides of the growth zone and thus reduces its ability to undergo tropism. The large decreases in magnitudes of φ_g, together with significant decreases in L_g, can explain why stage IVb sporangiophores from the two stiff mutant strains (C216 and C149) exhibit almost no tropic response.

The affect of a shorter growth zone on the bending rate is easily visualized. If the sporangiophore is capable of producing a constant bend rate per unit length of growth zone, then it follows that a longer growth zone will produce a larger bend rate compared to a shorter growth zone (Figure 7). This can explain how the shorter growth zones of the stiff mutant strains produce a smaller bend rate compared to WT.

The affect of a smaller irreversible wall extensibility of the growth zone (φ_g) is more difficult to visualize, but an example may help to explain how tropism may be reduced for stiff mutant sporangiophores. A WT stage IVb sporangiophore bends toward unilateral light at a rate of 1.5–3.0° min⁻¹ (Cerda-Olmedo and Lipson, 1987). If the average diameter of the sporangiophore’s growth zone is 120 μm, the distal side elongation rate of the growth zone must be approximately 6.0 μm min⁻¹ larger than the proximal side elongation rate to produce a bend rate of 3.0° min⁻¹. Using approximate values from Table 1, it can be determined using Eq. 12 that the proximal side elongation rate (dL/dt)_wp is 50 μm min⁻¹ when m_g is 1000 μm min⁻¹ MPa⁻¹ and (P – P_C) is 0.05 MPa. Using a value for L_g of 2000 μm, then φ_g is calculated to be 0.5 min⁻¹ MPa⁻¹. If φ_g is increased by 12% on the distal side then m_g is also increased by 12% (φ_g = 0.56 min⁻¹ MPa⁻¹ and m_g = 1120 μm min⁻¹ MPa⁻¹), and the distal side elongation rate (dL/dt)_wd = m_g (P – P_C) = (1120 μm min⁻¹ MPa⁻¹)·(0.05 MPa) = 56 μm min⁻¹. The difference between the distal and proximal elongation rate is (dL/dt)_d – (dL/dt)_p = 6.0 μm min⁻¹, producing a bend rate of 3.0° min⁻¹.

Now for a stiff mutant sporangiophore, the proximal side elongation rate (dL/dt)_sp is 50 μm min⁻¹ when m_g is 200 μm min⁻¹ MPa⁻¹ and (P – P_C) is 0.25 MPa. Using an average L_g of 1400 μm, then φ_g is calculated to be 0.143 min⁻¹ MPa⁻¹. Values used for φ_g, m_g, and (P – P_C) are approximate average values for C216 and C149 mutant strains (Table 1). Because φ_g is approximately 3.5 times smaller for the stiff mutants compared to WT, it may be assumed that the increase in φ_g on the distal side is also 3.5 times smaller. So if φ_g and m_g are increased by 3.4% on the distal side ((φ_g ≅ 0.149 min⁻¹ MPa⁻¹ and m_g ≅ 207 μm min⁻¹ MPa⁻¹), then the distal side elongation rate (dL/dt)_sd = m_g (P – P_C) = (207 μm min⁻¹ MPa⁻¹) (0.25 MPa) = 51.75 μm min⁻¹. The difference between the distal and proximal elongation rate is (dL/dt)_sd – (dL/dt)_sp = 1.75 μm min⁻¹, producing a bend rate of approximately 0.8° min⁻¹ which is significantly smaller than the bend rate of 3.0° min⁻¹ for the WT sporangiophore.

The difference in the mechanical properties of the cell wall for the WT sporangiophore and the stiff mutant sporangiophore can be illustrated with a dashpot element in series with two spring elements (Figure 8). The cell wall illustrations are similar to that in Figure 3B for a wall that exhibits tip growth. In Figure 8, the growth zone length and the irreversible wall extensibility (Bingham dashpot) are larger for WT sporangiophores’ cell walls (Figure 8A) compared to stiff mutant sporangiophores’ cell walls (Figure 8B). Also, the cell walls of the stiff mutant sporangiophores are under more tension than the WT as illustrated by the larger force arrows at the ends and the more extended springs in their wall (Figure 8B).

Additional insight into the biological processes affected in class-two mutant strains (stiff mutants) may be obtained. Prior research show that small vesicles within the cytoplasm move to areas of the wall where expansive growth occurs within cells with a growth zone, e.g., sporangiophores of P. blakesleeanus (Cerda-Olmedo and Lipson, 1987), fungal hyphae (Heath, 1990; Bartnicki-Garcia et al., 2000; Gierz and Bartnicki-Garcia, 2001), and pollen tubes and root hairs (Heath, 1990). It is thought that the small vesicles contain wall-building molecules that are released to the inner cell wall, via exocytosis, and mediate wall loosening and irreversible wall extension of the stressed wall (due to turgor pressure). Research conducted in this study demonstrates that magnitude of elongation growth rate depends on (a) the wall’s ability to extend irreversibly (i.e., the magnitude of φ_g), (b) the area of wall that extends irreversibly (i.e., the magnitude of L_g), and (c) the magnitude of the effective stress within the wall (i.e., the magnitude of P – P_C).

The changes in wall structure and/or architecture that produce the measured mechanical properties and elongation growth behavior of stiff mutant sporangiophores cell wall are unknown, but an examination of a few explanations may be insightful. It may be that the stiff mutant sporangiophores simply deliver more material to the wall making it thicker. Then the thicker wall would exhibit less irreversible extensibility (φ_g) as demonstrated in this study. However, it would be expected that the delivery of more material to the wall would also extend the length of the growth zone, which is inconsistent with the experimental results demonstrating that L_g of stiff mutant sporangiophores are actually shorter compared to WT. Another idea is that the delivered material makes more bonds with the existing wall material thus tightening and stiffening the wall more to make it less extensible. This idea suggests that the magnitude of φ_g is reduced for stiff mutants, but the growth zone length is not affected, which is inconsistent with the experimental results that demonstrate a decrease in L_g.

Still another idea is that a smaller amount of wall material is delivered to the inner wall of stiff mutant sporangiophores by transport vesicles and exocytosis. It is hypothesized that the new wall material catalyzes wall loosening in order to incorporate new material into the wall, using a process similar to that proposed for internode cells of C. corallina and plant cells (Boyer, 2009), and that m_g is a quantitative measure of the amount of wall loosening. So if the mass flow rate of wall-building molecules delivered to the inner cell wall is reduced, it follows that the magnitudes of m_g, φ_g, and L_g are reduced. Because the measured magnitudes of m_g, φ_g and L_g are significantly smaller for the stiff mutant sporangiophores tested, it can be deduced that the mass flow rates of wall-building molecules delivered to the inner cell wall are significantly smaller. Perhaps exocytosis is affected and/or the number of vesicles carrying wall-building molecules to the wall is smaller for stiff mutant sporangiophores from C216 and C149 strains. In conclusion it would appear that the last explanation is the most consistent with the experimental results. Therefore, it is hypothesized that the defective genes in stiff mutant sporangiophores (madD, E, F, G, and J) reduce the mass flow rate of wall-building materials (that catalyze wall loosening) to the inner wall thus reducing the magnitudes of m_g, φ_g, and L_g, and reducing their ability to perform tropic responses.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.