Framework for computational simulation of chemotropic growth

We developed a computational platform for studying yeast chemotropic growth. This platform is built on COMSOL with MATLAB (Comsol Inc.). Schematic diagrams of the simulation platform are given in . (i) The computational domain that contains the cells is taken to be a square. Initially, cells are assumed to have a circular geometry (). Biochemical species, reactions, boundary conditions, and initial conditions are defined in the relevant domains. (ii) We assume that cell growth occurs on a longer time scale than the chemical reactions and thermal diffusion. Therefore, all chemical species reach their steady state before cell growth occurs. This simplification allows us to separate solving the reaction-diffusion equations from growing the cells. Solving the reaction-diffusion equations produces the spatial profiles for the extracellular pheromone and Bar1 concentrations. (iii) The program determines the location of the maximum (*P*_{max}) and minimum (*P*_{min}) pheromone concentration around each cell. The relative gradient is then computed as 2(*P*_{max} − *P*_{min})/(*P*_{max} + *P*_{min}). If the absolute amount of pheromone and the relative gradient are above the threshold, a narrow rectangular segment is inserted between the half circle defined with *P*_{max} as its midpoint and the half circle defined with *P*_{back} as its midpoint (). Otherwise, the cell is elongated in a random or user-specified direction. The half circle, which contains *P*_{max} and is moved to accommodate the first inserted segment (S_{1}), is defined as the leading edge. The other half circle is taken to be the back of the cell and remains fixed throughout the simulation. (iv) After cell elongation, the program recomputes the steady-state profiles of the reaction-diffusion equations. (v) In later growth steps, only the pheromone concentration across the leading segment is considered, because this is where growth occurs. For this case, the relative gradient is computed as 2(*P*_{max} − *P*_{min})/(*P*_{max} + *P*_{min}), where *P*_{max} and *P*_{min} are the maximum and the minimum pheromone concentration, respectively, over the leading segment. If the relative gradient is above the threshold, a new segment (S_{n}
_{+ 1}) is inserted between the leading edge and the previous segment and rotated by an angle formed by the previous direction of growth and *P*_{max} (). Otherwise, the new segment is inserted in the same direction as the previous one. These last two steps are repeated until all cells grow in length to three times the original diameter or two cells collide with each other.

Chamber simulations

For the gradient simulations (), the concentration of pheromone on the left and right edges of the chamber domain is fixed at 0 and 100 nM, respectively, which creates a linear gradient across the chamber domain. In this case, no flux boundary conditions are used along the top and bottom edges of the chamber. For the chamber simulations with a constant pheromone background (), the pheromone concentration is fixed at all boundaries of the computational domain. The boundary conditions for Bar1 are absorbing at the four edges (the concentration is taken to be zero at these boundaries). The pheromone and Bar1 concentrations are computed from the following equations:

where

*D*_{bar1} and

*D*_{ph} are the diffusion coefficients for Bar1 and pheromone, respectively;

*s*_{b}(

*x*,

*y*) is the secretion of Bar1 from

*MAT***a** cells;

*d*_{b} and

*d*_{ph} are the rate constants for degradation of Bar1 and pheromone, respectively; and

*k*_{b} is the rate constant for Bar1-mediated degradation of pheromone. Initially,

*MAT***a** cells are assumed to release Bar1 at constant rate uniformly across the entire cell. Once the cells begin to elongate, Bar1 is released in a polarized fashion exclusively from the leading growth segment. The assumption of polarized release of Bar1 during chemotropic growth is not necessary to reproduce the qualitative features of the model. Pheromone-induced Bar1 production is simulated by increasing the flux out of the leading segment in proportion to the size of the cell.

Simulations with *MAT*a and *MAT*α cells

For simulations involving

*MAT***a** and

*MAT*α cells ( and ),

*MAT*α cells are assumed to produce α-factor uniformly across the cell surface. Bar1 is released from

*MAT***a** cells. The equation for Bar1 concentrations is the same as

Eq. 1, and that for pheromone is now given by

where

*s*_{ph}(

*x*,

*y*) models the uniform release of α-factor (75 nM/s) from

*MAT*α cells. The growth direction is determined using the relative gradient as described above. No flux boundary conditions (that is, no concentration is lost at the boundaries) are used for α-factor and Bar1.

Estimation of model parameters

There are seven model parameters: the synthesis and degradation rates of Bar1 and pheromone (*s*_{Bar1}, *d*_{Bar1}, *s*_{ph}, *d*_{ph}), the degradation of pheromone by Bar1 (*k*_{Bar1}), and the diffusion of pheromone and Bar1 (*D*_{ph}, *D*_{Bar1}). We estimated the synthesis, degradation, and diffusion rates of pheromone on the basis of the molecules’ size and the generation of a gradient that is 5 nM at the surface of a *MAT*α cell and drops to 20% of this value 100 μm away from the *MAT*α cell.

We assume that spatial gradients of pheromone and Bar1 exist in only two dimensions (

*x* and

*y*). This assumption is valid for the microfluidic chamber, which has a height of

*h* = 5 μm, and for the mating assays, which take place on an agar surface. The pheromone concentration

*C*_{p} is measured in units of nanomolar (nM). In the absence of other cells, the steady-state profile for

*C*_{p} around a

*MAT*α cell is described by the following equation:

where

is the Laplace operator in polar coordinates,

*d*_{P} is the pheromone degradation rate,

*D*_{p} is the diffusion coefficient, and

*j*_{P} is the pheromone flux density [molecules/(area-s)] at the cell boundary located at

*r*_{0}. The flux density

*j*_{P} is computed as follows. Assume pheromone molecules are synthesized inside the cell at a rate of

*s*_{ph} (nM/s) and released uniformly over the surface of the cell. Then, the flux per unit area is the product of synthesis rate and the ratio of the cell volume to surface area

*j*_{p} =

*s*_{ph}*r*_{0}/3.

The steady-state solution of

Eq. 4 is

where

*K*_{0} and

*K*_{1} are modified Bessel function of the second kind. The diffusion coefficient of pheromone is estimated from its molecular weight as 125 μm

^{2}/s. On the basis of

Eq. 5, we chose the synthesis rate of pheromone to be 75 nM/s and the degradation rate to be 0.005 s

^{−1}. With these values, the pheromone concentration is 5 nM at the surface of a

*MAT*α cell and drops to 20% of this value 100 μm away from the source. This synthesis rate requires a

*MAT*α cell with 5-μm diameter to produce 3000 pheromone molecules per second.

Bar1 emitted from a

*MAT***a** cell satisfies an equation analogous to

Eq. 4. Because Bar1 is substantially larger than pheromone, we chose its synthesis rate to be 1.5 nM/s, 50 times slower than that of pheromone. Once a

*MAT***a** cell begins to elongate, we assumed Bar1 is released exclusively from the leading edge. We assumed Bar1 degrades at a rate of 0.05 s

^{−1}, which is 10-fold faster compared with pheromone. We take the diffusion coefficient of Bar1 to be 6.25 μm

^{2}/s, which is 20 times slower than pheromone diffusion coefficient. With these values, the concentration of Bar1 in unit volume at the surface of the cell is 0.85 nM and drops to 20% of its value at a distance of 13 μm. The rationale for these choices is given next.

Parameter studies for self-avoidance of adjacent *MAT*a cells

Among the seven free parameters, we systematically varied *s*_{Bar1}, *d*_{Bar1}, *s*_{ph}, and *k*_{Bar1} and quantified their effects on self-avoidance and sharpening of the pheromone gradient.

For *MAT***a** cells to show self-avoidance (an angle of >20° between adjacent cells), the model parameters should satisfy two conditions: (i) Bar1-dependent degradation of pheromone (*k*_{Bar1}[Bar1]) needs to be ~2 orders of magnitude larger than spontaneous pheromone degradation (*d*_{ph}) (). This ratio can be increased by increasing the Bar1-related parameters *s*_{Bar1} and *k*_{Bar1}. Decreasing the Bar1 degradation rate *d*_{Bar1} also affects this ratio. However, this parameter increases the width of Bar1’s distribution around a *MAT***a** cell, which is in opposition to the second condition. (ii) We required that the distribution of Bar1 is localized around *MAT***a** cells (). If the Bar1 distribution is too broad, then, because of the first condition, the pheromone concentration around multiple *MAT***a** cells is too reduced, making the establishment of sharp pheromone gradients difficult and the angle between two neighboring *MAT***a** cells small. Two ways to restrict the Bar1 distribution are rapid degradation or slow diffusion. An additional mechanism that would ensure that Bar1 remains localized around *MAT***a** cells is if a portion of the protease remained trapped in the periplasmic space between the cell wall and the plasma membrane.

Parameter studies for sharpening pheromone gradients

Using the same geometry of *MAT***a** and *MAT*α cells as in , we investigated how the ability of Bar1 to sharpen pheromone gradients depended on the model parameters *s*_{Ph}, *d*_{Bar1}, and *k*_{Bar1}. The synthesis rate of pheromone, *s*_{Ph}, changed the amount of pheromone around a *MAT*α cell, but did not affect the relative gradient. Similar to improving self-avoidance, increasing the ratio of *k*_{Bar1}[Bar1]/*d*_{ph} sharpens pheromone gradients (). One way to increase this ratio is to decrease the Bar1 degradation rate *d*_{Bar1}, which broadens the Bar1 distribution and increases the amount of Bar1 in the medium. A uniform background of Bar1 is sufficient to generate large pheromone gradients, because under this condition, the α-factor distribution is proportional to

which asymptotically decreases as exp[−(

*k*_{Bar1}[Bar1]/

*D*_{p})

^{1/2}*r*] for large

*r* (

10). However, this sharpening of the gradient comes at the cost of a substantial reduction in the pheromone concentration, making it easy for the absolute amount of pheromone to drop below detection. Keeping the Bar1 concentration localized around

*MAT***a** cells allows α-factor concentrations to remain relatively high, while at the same time providing a mechanism for amplifying the pheromone gradient as

*MAT***a** cells elongate.

Yeast strains

A list of strains used in these studies and their complete genotype is provided in .

| **Table 2**Strains used in these studies. |

Chemotropic growth assays

The microfluidic device used for chemotropic assays and preparation of cells for imaging was described previously (

5). The pheromone concentration ranged from 0 to 100 nM in the cell chamber for differential interference contrast (DIC) and fluorescence imaging of

*BAR1* cells (BY4741-15;

*BEM1-GFP::His3MX6*) and 0 to 20 nM for

*bar1*Δ cells (BY4741-30;

*bar1*Δ

*::HisG-URA3-HisG BEM1-GFP::His3MX6*). To quantify self-avoidance for these two strains, we measured the angle between two adjacent

*MAT***a** cells every 50 min from the time that cells started to elongate during chemotropic growth. This angle is defined as the angle between the two lines from the growth tip to the contact point of the two cells. The alignment between the direction of growth and the gradient was quantified by the angle between vectors indicating the growth direction and the direction of gradient at 350 min. Microscopy was performed with a Nikon Ti-E inverted microscope equipped with a Photometrics CoolSNAP HQ2 Monochrome camera. Acquisition was performed with MetaMorph (Molecular Devices). Image processing and analysis was done with MATLAB (MathWorks) and ImageJ (

http://rsbweb.nih.gov/ij/).

Quantitative mating assays

*BAR1* and

*bar1*Δ strains with nonrevertible and complementing nutritional markers were derived from BY4741 and BY4742 () and assessed for opposite cell type (

*MAT***a** ×

*MAT*α) and same cell type (

*MAT***a** ×

*MAT***a**) mating efficiency in the absence or presence of exogenous pheromone, using a modification of the procedure described by Sprague (

20). In these experiments, mating mixtures (, crosses 1 to 4) were made with 1 × 10

^{6} cells of each mating partner in suspensions (200 μl) containing 0, 0.75, 1.5, 3, or 6 μM exogenous mating pheromone (α-factor). Each suspension was pipetted onto a 25-mm filter (0.45-μm pore size; Millipore Corp.) on the surface of a separate YPD (yeast extract, peptone, and dextrose) plate with the corresponding concentration of mating pheromone. Cells for these mixtures were grown in liquid YPD medium to the early log phase (5 × 10

^{6} to 1.5 × 10

^{7} cells/ml). After 5 hours (30°C), the cells were collected from the filters and diluted for plating on selective medium (synthetic dextrose supplemented with histidine, leucine, and uracil) to determine diploids per milliliter, and on nonselective medium (synthetic complete dextrose) to determine total cells per milliliter (diploids and haploids). Reported mating efficiencies are the ratio of diploids to total cells normalized to that for the reference

*MAT***a** ×

*MAT*α

*BAR1* mixture without pheromone (, cross 2). Three independent assays were done for each mating mixture at the specified pheromone concentrations.

In experiments to assess the effects of local Bar1, *MAT***a**
*BAR1* or *bar1*Δ equalizer cells were included in the mixtures with mating partners that have complementing selectable markers (, crosses 9 to 12). *MAT***a** × *MAT***a**
*BAR1* or *bar1*Δ mating mixtures were made with 2.5 × 10^{5} cells of each mating partner and 5 × 10^{5} cells of the *MAT***a**
*bar1*Δ or *BAR1* equalizer strain, respectively. Similarly, *MAT***a** × *MAT*α *BAR1* or *bar1*Δ mating mixtures were made with 2.5 × 10^{5} cells of each mating partner and 2.5 × 10^{5} cells of the *MAT***a**
*bar1*Δ or *BAR1* equalizer strain. (Note that *MAT*α *BAR1* cells do not produce or secrete Bar1.) Cell cultures and mating mixtures were prepared and plated to quantify mating events as described above. Mating efficiencies for these comparisons (, crosses 9 to 12) are the ratio of diploids to total cells normalized to the *MAT***a** × *MAT*α *BAR1* mixture containing *bar1*Δ equalizer cells (, cross 10). Three independent assays were done for each mating mixture.

The auxotrophic markers in these strains are coding sequence deletions that are nonrevertible. Therefore, only fusion products with complementing nutritional markers (*LYS2/lys2*Δ_{0} met15Δ_{0}/MET15) grow on selective medium. To confirm this assertion, we included two plating controls in parallel with the quantitative mating assays. First, 2 × 10^{6} cells of the *MAT***a** haploid strains used for mating mixtures 1 and 3 () were incubated separately on YPD filters and plated on selective medium at the same dilution as for the *MAT***a** × *MAT***a** mating mixtures. Second, crosses between opposite cell-type partners with noncomplementing selectable markers (, crosses 5 to 8) were made in the absence of exogenous pheromone, incubated on YPD filters, and plated on selective medium at the same dilution as for the *MAT***a** × *MAT*α mating mixtures. No colonies were observed on any of these control plates.

Cell-type verification

The cell type of rare diploids from

*MAT***a** ×

*MAT***a** mating mixtures was tested to discern whether they are products of same or opposite cell type mating. (The latter could result from mating-type switching in rare cells in the population, a process that involves replacing the

**a** or α allele at the

*MAT* locus with sequence from the

*HML*α or

*HMR***a** locus, respectively.)

*MAT***a**/

*MAT***a** diploid cells mate efficiently with

*MAT*α but not with

*MAT***a** haploids to form viable triploid fusion products. By contrast,

*MAT***a**/

*MAT*α diploid cells mate with neither. We performed a qualitative mating assay with tester strains KZ8-5C (

*MAT***a**) and KZ8-1D (

*MAT*α) to test isolated colonies from the selective plates to distinguish between

*MAT***a**/

*MAT***a** and

*MAT***a**/

*MAT*α fusion products (

19). Isolates from selective plates corresponding to

*MAT***a** ×

*MAT***a** mating mixtures made without or with the specified amounts of exogenous pheromone were tested for mating ability (200

*BAR1* and 200

*bar1*Δ isolates from two independent sets of experiments). Forty isolates from the

*BAR1 MAT***a** ×

*MAT*α mating mixture without exogenous pheromone were included for reference. All 400 diploids that were tested from the

*MAT***a** ×

*MAT***a** mixtures mated with the

*MAT*α tester but not the

*MAT***a** tester strain, consistent with assignment of the

*MAT***a** cell type to these fusion products. As expected, none of the 40

*MAT***a**/

*MAT*α diploids mated with either tester strain. Additionally, 12 isolates each from selective plates of the

*MAT***a** ×

*MAT***a** and

*MAT***a** ×

*MAT*α mixtures without exogenous pheromone were tested for sporulation. No spore asci were observed for products from the same-sex mating mixtures, whereas all of those from the opposite mating-type mixtures produced spore asci.