An Inverse Modified Helmholtz

Problem for Identifying Morphogen

Sources from Sliced Biomedical Image

Data

Marcus PensaSupervised by Mark Flegg

Monash University

February 26, 2018

Vacation Research Scholarships are funded jointly by the Department of Education and Training

and the Australian Mathematical Sciences Institute.

1 Introduction

During both female meiosis (cell division) I and II, an oocyte (egg cell) divides asymmetrically resulting in the

production of a small cell, called a polar body, which ultimately breaks down [Schmerler and Wessel, 2011].

An important early event in this division is asymmetrical accumulation of a molecular polymer called actin at

the oocyte cortex, where the greatest actin accumulation occurs at a location which coincides with the location

of polar body formation and extrusion [Longo and Chen, 1985]. The actin thickening forms adjacent to the

meiotic spindle, a structure which helps separate the cell’s chromosomes into a set(s) which will stay in the

oocyte, and a set(s) which will enter the polar body. It is known that DNA, a molecule which is a constituent

part of a chromosome, can act at a distance to induce the formation of the actin cap, however, the way in which

the DNA communicates with the cell surface remains unknown [Deng et al., 2007].

In attempt to explain the observed actin thickening, we hypothesis that a morphogen (diffusible signalling

molecule) is produced with an asymmetrical distribution in the oocyte, possibly from a location which coincides

with the location of DNA in the cell, and that the morphogen promotes actin formation when it reaches the

cell membrane. Furthermore, we hypothesise that the morphogen is processed or broken down upon reaching

the cell membrane. In order to create a model for our hypothesis, we assume that morphogen is generated from

one or more point sources at a fixed (but to be determined) location in the oocyte and that the morphogen

generation, decay as well as diffusion rates are also fixed parameters which are yet to be determined. We assume

that the concentration of morphogen is zero at the cell surface, due to processing of the morphogen at the cell

surface, and that morphogen levels reach equilibrium. The mathematical model of morphogen concentration

is represented by an inhomogeneous modified Helmholtz equation with zero boundary conditions. We assume

that actin thickness is in pseudo-equilibrium and that actin thickness depends only on a linear production-

degradation process whereby production is caused by signalling from the putative morphogen. Equivalently,

these assumptions can be stated as actin thickness being proportional to the flux (amount broken down) of

morphogen through the cell membrane.

This model is motivated by a desire to determine unknown parameters and location of production for

the putative morphogen where the available data for analysis is the distribution of cortical actin. Therefore,

algorithms were generated which solve the inverse problem; the algorithms determines what model parameters

should be used such that the the measured/known actin distribution is the outcome.

To solve the inverse problem, Duxbury [2016] has previously presented a least square approach which employs

a look-up table. In this report, we describe an iterative approach that does not require interpolation of a look-up

table and still uses a least square method. Furthermore, we present a weighted least squared approach.

Using our model, we were able to generate control data by finding theoretical actin distributions for sets

of possible parameters and sampling the resultant actin distribution on concentric rings of the cell surface,

then adding noise. This control data emulates real data which is available from confocal microscopy images of

oocytes where actin thickness is visible in each slice image. Our un-weighted least square inverse algorithm was

validated using this control data.

1

2 Model

Let u denote the morphogen concentration at any point in space r ∈ Ω, as described using spherical co-ordinates

where θ is the azimuthal angle; r = (r, θ, φ). We will assume a spherical cell volume, Ω, centred at the origin,

with radius R. Let the surface of the cell be given by ∂Ω, and because moprhogen molecules striking the cell

surface are absorbed, u = 0 on ∂Ω. We model the transport of morphogen using the continuity equation with a

generation term, f(r), and with a degradation term, µu, where µ represents break down rate of the morphogen.

By assuming our system reaches equilibrium and by applying Fick’s first law, the continuity equation with our

boundary condition results in the BVP

−D∇2u(r) + µu(r) = f(r), r ∈ Ω (1a)

u(r) = 0, r ∈ ∂Ω (1b)

where D is the diffusivity of the morphogen.

We can non-dimensionalise the BVP given by (1a) and (1b) by rescaling Ω isotropically by a factor of R.

Using this rescaling, the BVP becomes

−D∇2u(r) +R2µu(r) = R2f(r), r ∈ Ω (2a)

u(r) = 0, r ∈ ∂Ω (2b)

where our new domain, Ω, is the origin-centred unit sphere, r = rR and r = r

R . To express the DE (2a) in the

form of a standard inhomogeneous Helmholtz equation, we will divide through by D and introduce κ = R2µ/D

as well as F (r) = R2f(r)/D:

−∇2u(r) + κ2u(r) = F (r), r ∈ Ω (3a)

u(r) = 0, r ∈ ∂Ω (3b)

.

To solve the BVP given by (3a) and (3b), we introduce the Greens function, G, defined such that

−∇2G(r; r0) + κ2G(r; r0) = δ3(r− r0), r ∈ Ω (4a)

G(r) = 0, r ∈ ∂Ω (4b)

where r0 = (r0, θ0, φ0).

The solution to the BVP described by (4a) and (4b) is

G(r; r0) =κ

2π2

∞∑l=0

(2l + 1)

(il(κr<)kl(κr>)− kl(κ)il(κr0)il(κr)

il(κ)

)Pl(cosφ′0), 0 ≤ r ≤ 1 (5)

Where Pl are Legendre polynomials, il and kl are Modified Spherical Bessel functions of the first and second

kind, r< = min(r, r0), r> = max(r, r0) and

cos(φ′i) = cos θ sinφ cos θi sinφi + sin θ sinφ sin θi sinφi + cosφ cosφi.

2

See the Appendix for a derivation of this solution.

By construction of the Greens function, we can express u in terms of the Greens function as

u =

Ω

G(r; r0)F (r0) dr0. (6)

We can model the case where morphogen generation occurs from n point sources by setting F (r) such that

F (r) =

n∑i=1

λiδ3(r− ri).

By substituting this definition of the function F into equation (6), we can write u as a linear combination of

Green’s functions:

u =

n∑i=1

λiG(r; ri). (7)

If we let a(θ, φ) represent the thickness of actin on the cell boundary for a unit sphere cell and assume that

actin generation is proportional to morphogen flux through the cell boundary, then we can model the rate of

actin generation by the equation∂a

∂t= κa∇u · n |r=1 − µaa

where n is an outward pointing normal unit vector on the unit sphere, κa is a constant of actin generation rate

and µa is a constant of actin breakdown rate.

However, the outward pointing unit normal vector of a sphere is r, and the flux in the radial outward

direction can alternatively be expressed as the rate of change with respect to radius. Hence we can express the

rate of actin thickness change with respect to time as

∂a

∂t= κa

∂u

∂r

∣∣∣∣r=1

− µaa. (8)

If we assume that actin thickness is in pseudo-equilibrium, then we can re-write equation (8) as

a ∝ ∂u

∂r

∣∣∣∣r=1

. (9)

Recalling both (5) and (7), the actin distribution is given by

a ∝n∑i=1

λi∂G(r; ri)

∂r

∣∣∣∣r=1

=

n∑i=1

λi

(κ2

2π2

∞∑l=0

(2l + 1)

(il(κri)k

′l(κ)− kl(κ)il(κri)i

′l(κ)

il(κ)

)Pl(cosφ′i)

). (10)

We can simplify this expression by using Wronskian identity i′l(κ)kl(κ) = π/(2κ2) + il(κ)k′l(κ):

a ∝ 1

4π

n∑i=1

(λi

∞∑l=0

(2l + 1)

(il(κri)

il(κ)Pl(cosφ′i)

)). (11)

Relationship (11) gives us a compact way to express theoretical actin thickness in terms of parameters

relating to normalised degradation (κ), morphogen production intensity (λ) and source locators (ri, θi, φi) of

the putative morphogen.

3

2.1 Modelling a single point source

Consider the case when the generation of morphogen occurs only as a single point source. Then relation (11)

becomes

a ∝ 1

4π

∞∑l=0

(2l + 1)

(il(κr1)

il(κ)Pl(cosφ′1)

). (12)

It should also be noted that calculating l consecutive Legendre polynomials using Matlab’s inbuilt Legendre

polynomial function is slow. It is faster to use the recurrence relationship

Pl+1 =(2l + 1) cosφ′1Pl − lPl−1

l + 1.

Because all Legendre polynomials over the interval [−1, 1] have a magnitude with an upper bound of one, the

best point to truncate the series such that (12) has a small error is dependent on the Modified Spherical Bessel

quotientil(κr1)

il(κ).

The code used to generate the algorithms discussed in this report truncate the series when the lth Bessel quotient

is smaller than a one millionth of the first Bessel quotient. For extreme values of κ where κ→ 0,∞ or r1 where

r1 → 0, 1, Matlab is unable to calculate many terms of the Bessel quotient. This is because both il(κr1) and

il(κ) are individually extremely large or small, even though the quotient is well behaved. Fortunately, a code

does not need to be generated which can evaluate equation (11) for the cases when κ or r1 are large. This is

because a large κ value means that the morphogen is broken down much faster than it diffuses and a large r1

value means that the point source of morphogen production is very close to the cell surface. Either of these

two cases would result in a very sharp peak of actin on the membrane which by inspection of the experimental

data is clearly not the case. For the cases when κ or r1 are small, it is useful to consider the following bounds

on the Bessel quotient [Paris, 1984]:

rl1 expκ(r1−1) <il(κr1)

il(κ)< rl1, (13)

where the upper bound also approximates the long term behaviour for κ l. For the case when r1 is small, the

Bessel quotient needs to be evaluated for few terms as the Bessel quotient is dominated by rl1 which will quickly

approach zero. For the case when κ is small, the bound becomes very tight and rl1 quickly becomes a good

approximation of the Bessel quotient. The algorithms discussed in this report begin approximating the Bessel

quotient using the long term behaviour approximation after one of two cases: (1) the difference between the long

term behaviour approximation and Bessel quotient is less than a one millionth of the size of the approximation

or (2) the difference between the long term behaviour approximation and Bessel quotient is less than a one

thousandth of the size of the approximation and the lth Bessel quotient is smaller than a one thousandth of the

first Bessel quotient.

4

3 Experimental data

We have obtained images in 2-D cross sections, spaced 2µ and oriented perpendicular to the z-axis, of mouse

oocytes undergoing their second asymmetrical division. The z-stack images are taken without prior knowledge

of the orientation of the actin cap or the adjacent spindle structures, however, a sufficient number of images are

taken to sample the entire spindle. The series of images begin before the spindle enters the field of view and

end after the spindle leave the field of view. Because the actin cap is adjacent to the spindles, the collection of

images are centred about the location of the cell where actin is thickest. The number of images available per

cell ranges from 14 to 24. Given that a typical oocyte being analysed has a diameter of approximately 70µm,

the amount of the oocyte being sampled from these images ranges from approximately 37% (26/70) to 66 %

(46/70). Figure 1 shows a sample of images from a series of 21 successive z-stack images. The actin can be seen

in green and the mitotic spindle in red and blue. The images are displayed only for some slices; (a) z0, (b) z7

(14µm above z0), (c) z17 (34µm above z0) and (d) z21 (42µm above z0).

(a) z0. (b) z7. (c) z17. (d) z21.

Figure 1: Confocal microscopy images of meiosis II oocytes showing cortical actin (green) and the meiotic

spindle (chromosomes in blue, microtubules in red). Image supplied by Dr Wai Shan Yuen.

Notice in Figure 1 that the actin is thickest adjacent to the meiotic spindle. Also, in the slice z21 a polar

body formed from meiosis I is visible. Discriminating between the actin from the polar body and actin from the

egg cell is a challenge which we are yet to resolve. A possible solution is to analyse meiosis I oocytes as these

cells have not yet extruded a polar body.

4 Control data

Control data was generated to validate an algorithm which attempts to match actual data to the closest

theoretical distribution. The control data was generated to emulate the experimental data where approximately

half of the cell is sampled with evenly spaced z stack images, and the slices are centred about the thickest region

of actin in the cell. For a set of possible parameters, r1, θ1, φ1, κ and λ the forward solution is calculated using

equation (12) giving a theoretical actin distribution. Centring the emulated data around the thickest region of

actin in the cell was achieved using the fact that if there is a single point source of signalling molecule, then peak

5

actin intensity on the surface theoretically occurs at the same orientation as the point source relative to the

origin. For a theoretical 70µm cell with an actin distribution centred at φ = 1, slices were taken such that they

were approximately centred about φ = 1. Seventeen slices, each 2µm apart on the z axis, were taken beginning

from the middle of the cell and the final slice was 3µm from the top of the cell. On each slice, actin intensity

was sampled at 20 evenly spaced θ values. Noise was added both multiplicatively and additively at each data

point, j, to the actin data, aj , using the formula

aj = aj + ζMaj + ζA

where aj is control data with simulated noise at a given data point and ζ ∼ N(0, 12 expPk/5) for k = M,A.

We call Pk the noise power. Furthermore, because actin thickness is non-negative, if adding noise results in

aj < 0, then we set aj = 0. Figure 2 illustrates typical control data.

Figure 2: Actin distribution for a typical control problem

5 Inverse problem and results

The inverse problem involves finding parameters r1, φ1, θ1, κ and λ which lead to an actin distribution which

best matches the data. Due to φ1 and θ1 being the most robust parameters and able to be independently

evaluated using a centre of mass approach, as presented by Duxbury [2016], we generated an inverse algorithm

which requires known θ1 and φ1 as inputs. Thus the inverse problem becomes a 3-parameter optimisation in

r1, κ and λ.

6

5.1 Un-weighted least square approach

For the un-weighted approach, an algorithm was created which uses the sum of square residuals as an objective

function; a least square regression was used. Matlab’s fminsearchbnd function, which uses a Nelder-Mead

method, was used to minimise the objective function. Additionally, this function allows bounds to be placed on

parameters, thus preventing impossible parameter spaces from being interrogated (eg κ < 0) and the algorithm

breaking down. This approach is similar to that presented by Duxbury [2016], however, the new approach

notably differs in that calculation of a look-up table is not required. When applied to noise free data the new

approach returns actin distributions and parameters that fit the control problem with very high accuracy and

precision. This is an improvement on the previous method which in some cases returned a bad fit. Figure 3

illustrates a case when the look-up table approach failed to return a good fit while the new approach succeeded.

(a) Solution using look-up table method. (b) Solution using new approach.

Figure 3: Comparison of the least square optimised solution returned using an algorithm with and without a

look-up table

Notice that in Figure 3 (b) the sampled vales of φ′1 do not extend to π due to the fact the new algorithm

was tested using control data where data was only collected on slices spaced such that they do not span the

entire cell.

The un-weighted least square algorithm was also tested on noise containing control data. More specifically,

the algorithm was run on on 5 sets of 50 control problems where each set has a different level of noise power

but the same set of parameters such that r1 = 0.5, φ1 = 1, θ1 = 3, κ = 2 and λ = 1. The errors returned by

the algorithm for each of the parameters (a) r1, (b) κ and (c) λ are presented in Figure 4. For each parameter,

when the noise level was low the average error is close to zero, however, with increasing noise the average error

deviated from zero in a way which appears to be systematic. This change in average error may be due to noise

not being normally distributed which is a result of rejecting a proportion of noise if it resulted in aj becoming

7

negative.

(a) (b)

(c)

Figure 4: Comparison of the least square optimised solution returned using an algorithm with and without a

look-up table

5.2 Weighted least square

For the weighted approach, an algorithm was created which uses the weighted sum of square residuals as

an objective function where the weighting is proportional to the theoretical actin distribution’s gradient at

the interrogated data point. It was decided to heavily weigh data points corresponding to regions on the

distribution where the gradient is steep because the shape of the distribution is determined most strongly by

regions of the steepest gradient. The gradient was calculated using the gradient formula where the two actin

intensities selected corresponded to (1) the φ′1 value of the data point and (2) the φ′1 value which is one degree

(1) larger.

8

This algorithm returned parameters with very high precision and accuracy for noise free control data,

however, each iteration is slower and many more iterations are required using this approach relative to the

un-weighted algorithm. Furthermore, this method returned large errors for all three parameters when being

run on control problems with noise.

6 Discussion

The un-weighted algorithm performed significantly better than the weighted algorithm. Of the three parameters

being optimised, r1, κ and λ, the most robust (able to be returned most accurately) parameter was r1 while

large error was associated with the κ and λ values returned. Thus, for control data our un-weighted algorithm

is well suited to locating the theoretical point source of morphogen production, but not describing morphogen

characteristics (modified breakdown rate or production intensity). The instability of the 3-parameter optimi-

sation exists because very similar actin distributions can result from significantly different parameters. Moving

the point source of morphogen closer to the cell surface, corresponding to ↑ r1, or decreasing the modified

breakdown rate, ↓ κ, or increasing source intensity, ↑ λ, can all increase the amplitude of the distribution in

such a way that change in one variable can be mostly compensated by changes in the other two variables. Figure

5 shows two similar curves corresponding to vastly different parameters.

Figure 5: Similar actin distributions returned for different parameters

A two parameter optimisation of κ and λ should be more reliable as of these two parameters only κ can

effect the shape of the actin distribution. A different limitation of the method we present is that the morphogen

production source is likely from each chromosome rather than from a single point source in the cell (as assumed).

Fortunately, a biological method exists where the chromosomes can be removed from the cell and a glass bead

coated in DNA can be injected and ectopically induce an actin thickening [Deng et al., 2007]. Using data

generated form this approach could allow us to control the location of morophogen production and limit this

9

location to a small volume of the cell, the glass bead, resembling a point source. Furthermore, having a known

location of the morphogen point source could reduce our problem to a two parameter optimisation, therefore

removing the instability associated with the 3 parameter approach and also removing the need to apply a centre

of mass approach to find φ1 and θ1.

7 Conclusion

This report has vigorously developed the mathematics to find the theoretical actin distribution on the oocyte

surface under the assumption that there are any finite number of point sources of morphogen production. For

the case where there is a single point source located at a known angle from the centre of the cell, a 3-parameter

optimisation algorithm was generated which can identify the most likely point source location (radius) as well

as the modified breakdown rate and production intensity of the morphogen. Based on testing with control

data, the error associated with the predicted location (radius) of the point source is small while large error is

associated with the predicted modified breakdown rate and production intensity of the morphogen.

8 Appendix - Deriving the Greens function solution

To find the solution to BVP described by (4a) and (4b), we will transform the BVP using a transformation

mapping r : (r, φ, θ) → r′0 : (r′0, φ′0, θ′0) where the transformation is a rotation such that r′00 sits on the new

z axis as given by φ′0 = 0. By performing this transformation, we simplify the problem so that the solution is

independent of θ′0. The angle from the new z axis to any other point in the domain can be described by

cosφ′0 = (r′0 · r0)/(|r′0||r0|) = cos θ sinφ cos θ0 sinφ0 + sin θ sinφ sin θ0 sinφ0 + cosφ cosφ0

The new radius is given by r′0 = r. Because they are interchangeable, and due to simplicity of notion, we will

avoid using r′0 and instead use r where possible. In our new co-ordinate system, the PDE (4a) can be written

as

−∇2G+ κ2G = δ3(r′0 − r0′0) (14)

Consider equation (14) when r′0 6= r0′0. In this case, equation (14) reduces to the homogeneous Helmholtz

equation which we can solve using a separation of variables approach. Lets look for solutions of the form

G = R(r)Φ(φ′0). Substituting this form of solution into the homogeneous Helmholtz equation gives

−(

Φ

r2

∂

∂r

(r2 ∂G

∂r

)+

R

r2 sinφ′0∂

∂φ′0

(sinφ′0

∂Φ

∂φ′0

))+ κ2RΦ = 0

Notice that the θ′0 dependent term in the Laplacian were omitted and this is because the solution is

independent of θ′0, so the θ′0 dependent term vanish.

Divide through by RΦ/r2, then move all r dependent terms to RHS and keep φ′0 dependent terms on LHS:

− 1

Φ sinφ′0∂

∂φ′0

(sinφ′0

∂Φ

∂φ′0

)=

1

R

∂

∂r

(r2 ∂G

∂r

)− κ2r2

10

Because LHS is independent of r, and RHS is independent of φ′0, but RHS=LHS, then both LHS and RHS

must each be independent of both r and φ′0. That is, we can set both sides of the equation equal to a constant.

In particular, we will choose the constant l(l + 1) for l = 0, 1, 2, 3... This process yields two equations for us to

solve:

1

R

∂

∂r

(r2 ∂G

∂r

)− κ2r2 = l(l − 1) (R)

AND

− 1

Φ sinφ′0∂

∂φ′0

(sinφ′0

∂Φ

∂φ′0

)= l(l − 1) (Φ)

8.1 Solving equation (R)

We can nondimensionalise (R) via rescaling by a factor of κ. Using this rescaling, then moving all terms to

LHS, (R) becomes the Modified Spherical Bessel Differential Equation

τ∂2R

∂τ2+ 2τ

∂R

∂τ− (τ2 + l(l + 1))R = 0

where τ = κr. This has the general solution

Rl = Alil(τ) + Blkl(τ) = Alil(κr) + Blkl(κr) (15)

Where Al and Bl are real constants, il and kl are respectively Modified Spherical Bessel functions of the first

and second kind.

8.2 Solving equation (Φ)

Apply the product rule to equation (Φ), multiply through by −Φ, expand and move all terms to LHS so that

the equation in a from that is recognisable as the Legendre DE:

∂2Φ

∂(φ′0)2+

cosφ′0

sinφ′0∂Φ

∂φ′0+ Φl(l + 1) = 0

This has the general solution

Φl = ClPl(cosφ′0) +DlQl(cosφ′0)

Where Cl and Dl are real constants, Pl and Ql are respectively Legendre polynomials of the first and second

kind. However, we require that our solution is bounded, so we must have Dl = 0 ∀l. Thus

Φl = ClPl(cosφ′0) (16)

8.3 Putting it together

We have now found Rl and Φl, so we can describe all solutions of the form Gl = RlΦl. Furthermore, by the

principle of superposition, a general solution can can be described as the sum of all Gl solutions:

11

G =

∞∑l=0

(Alil(κr) +Blkl(κr))Pl(cosφ′0) (17)

where Al = AlCl and Bl = BlCl

8.4 Determining Al and Bl

Lets divide our solution into a component where ro < r and r < r0 by introducing G±, g±l , A±l and B±l such

that:

G+ =

∞∑l=0

g+l Pl(cosφ′0) =

∞∑l=0

(A+l il(κr) +B+

l kl(κr))Pl(cosφ′0), r0 ≤ r ≤ 1 (18)

G− =

∞∑l=0

g−l Pl(cosφ′0) =

∞∑l=0

(A−l il(κr) +B−l kl(κr))Pl(cosφ′0), 0 ≤ r ≤ r0 (19)

We have four unknowns, A±l and B±l , but only 3 equations, two boundary conditions and the continuity of

Green’s functions. To obtain a fourth equation lets look at the Green’s function ”jump” condition.

8.5 Green’s function jump condition

Multiply equation (14) through by Pn(cosφ′0), where Pn is the nth Legendre polynomial. Then integrate with

respect to r′0 over the origin centred infinitesimally thin spherical shell with radius r which we will denote as

Ωr0

−

Ωr0

Pn(cosφ′0)∇2Gdr +

Ωr0

Pn(cosφ′0)κ2Gdr =

Ωr0

Pn(cosφ′0)δ3(r′0 − r0′0)dr (20)

Using the fact that Pn∇2G = ∇ · (Pn∇G) − ∇Pn∇G, we can decompose the first integral into two further

integrals, one of which vanishes because we are integrating a finite function over an infinitesimally small domain.

We will apply divergence theorem to the remaining integral coming out of the decomposition. The second integral

from equation (20) also vanishes due to being an integration of a finite function over an infinitesimally small

domain. Using properties of the dirac delta function, we can evaluate the LHS integral. Therefore, Equation

(20) becomes

−

Ωr0

∇(Pn(cosφ′0)∇G) · ndr = Pn(cosφ′00 ) (21)

where n is an outward pointing normal unit vector on the shell Ωr0 . On the outside of the shell, which we shall

denote as Ω+r0 , the normal is given by n = r′0. On the inside of the shell, Ω−r0 , n = −r′0. Notice that

∇(Pn(cosφ′0)∇G) · ±r

is simply a scaled directional derivative of G in the inward or outward direction, so can alternatively be expressed

as a scaled partial derivative of G with respect change in radius. Also recall that φ′0 = 0. We can therefore

12

write equation (21) as

Ω+r0

∂G+

∂r

∣∣∣∣r=r0

Pn(cosφ′0)dr−

Ω−r0

∂G−

∂r

∣∣∣∣r=ro

Pn(cosφ′0)dr = −Pn(cos 0) = −1 (22)

Using the definition of G± from equations (18) and (19) in equation (22), then factorising, gives

∞∑l=0

((∂g+

l

∂r

∣∣∣∣r=ro

−∂g−l∂r

∣∣∣∣r=ro

)Ωr0

Pl(cosφ′0)Pn(cosφ′0)dr

)= −1

Integrating in spherical co-ordinates gives

∞∑l=0

((∂g+

l

∂r

∣∣∣∣r=ro

−∂g−l∂r

∣∣∣∣r=ro

) 2π

0

π

0

Pl(cosφ′0)Pn(cosφ′0)r20 sinφ′0dφ′0dθ′0

)= −1

Notice that the expression being integrated is independent of θ′0. Perform a the substitution x = cosφ′0.

Under this change of variables, the equation becomes

−1 = 2πr20

∞∑l=0

(∂g+

l

∂r

∣∣∣∣r=ro

−∂g−l∂r

∣∣∣∣r=ro

) 1

−1

Pl(x)Pn(x)dx

Using orthogonality of Legendre polynomials and rearranging the equation gives the jump condition

−2l + 1

4πr20

=∂g+

l

∂r

∣∣∣∣r=ro

−∂g−l∂r

∣∣∣∣r=ro

(23)

8.6 Using the boundary conditions

Because our solution must be bounded at the origin, we require that B−l = 0. Enforcing the condition (4b),

which can alternatively be expressed as G(1, θ′0φ′0) = 0, gives

B+l = −A+

l

il(κ)

kl(κ)

Using this definition of B+l and the fact that A−l = 0, we can write

g−l = A−l il(κr) (24)

AND

g+l = A+

l

(il(κr)−

il(κ)kl(κr)

kl(κ)

)(25)

8.7 Using continuity of the Greens function

By definition, the Green’s function is continuous, so g+l = g−l . Equating g+

l and g−l from equations (24) and

(25), then solving for A−l yields

A−l = A+l

(1− il(κ)kl(κr0)

kl(κ)il(κr0)

)(26)

As seen in equation (25), g+l can be expressed with A+

l as the only unknowon. We can now express also

express g−l where the only unknown is A+l :

g−l = A+l

(1− il(κ)kl(κr0)

kl(κ)il(κro)

)il(κr) (27)

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If we differentiate (24) and (25) with respect to r at r0, then use these definitions in the jump condition as

described by equation (23) we get:

− (2l + 1)

4πr20

= κA+l

(il(κr)−

il(κ)kl(κr)

kl(κ)

)− κA+

l

(1− il(κ)kl(κr0)

kl(κ)il(κro)

)il(κr) (28)

Solving equation (28) for A+l gives

A+l =

(2l + 1)il(κr0)kl(κ)

4πr2oκil(κ)

∗ 1

il(κr0)k′l(κr0)− i′l(κr0)kl(κr0)

Using the definition of the Wronskian for Modified Spherical Bessel functions, we can simplify our definition of

A+l to

A+l = − (2l + 1)il(κr0)kl(κ)κ

2π2il(κ)(29)

Using this definition A+l in equations (25) and (27), then rearranging RHS for each equation yields

g+l =

(2l + 1)κ

2π2

(il(κr0)kl(κr)−

kl(κ)il(κr0)il(κr)

il(κ)

), r0 ≤ r ≤ 1 (30)

AND

g−l =(2l + 1)κ

2π2

(il(κr)kl(κr0)− kl(κ)il(κr0)il(κr)

il(κ)

), 0 ≤ r ≤ r0 (31)

Using these definitions of g+l and g−l in equations (18) and (19) we can rewrite G+ and G− as

G+ =

∞∑l=0

(2l + 1)κ

2π2

(il(κr0)kl(κr)−

kl(κ)il(κr0)il(κr)

il(κ)

)Pl(cosφ′0), r0 ≤ r ≤ 1 (32)

AND

G− =

∞∑l=0

(2l + 1)κ

2π2

(il(κr)kl(κr0)− kl(κ)il(κr0)il(κr)

il(κ)

)Pl(cosφ′0), 0 ≤ r ≤ r0 (33)

We can alternatively express G as

G =

∞∑l=0

(2l + 1)κ

2π2

(il(κr<)kl(κr>)− kl(κ)il(κr0)il(κr)

il(κ)

)Pl(cosφ′0), 0 ≤ r ≤ 1 (34)

where r< = min(r, r0) and r> = max(r, r0)

References

Manqi Deng, Praveen Suraneni, Richard M Schultz, and Rong Li. The ran gtpase mediates chromatin signaling

to control cortical polarity during polar body extrusion in mouse oocytes. Developmental cell, 12(2):301–308,

2007.

K Duxbury. Reaction diffusion model for human oocyte symmetry breaking. Technical report, AMSI, 2016.

Frank J Longo and Da-Yuan Chen. Development of cortical polarity in mouse eggs: involvement of the meiotic

apparatus. Developmental biology, 107(2):382–394, 1985.

14

RB Paris. An inequality for the bessel function j ν(νx). SIAM journal on mathematical analysis, 15(1):203–205,

1984.

Samuel Schmerler and Gary M Wessel. Polar bodies - more a lack of understanding than a lack of respect.

Molecular reproduction and development, 78(1):3–8, 2011.

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