Appendix A
A general N2-fixing cell model (spherical geometry)
We consider a spherical coordinate system, where r denotes the distance from the
origin (i.e. the centre of the heterocyst). The general N2-fixing cell itself is idealized
as a spherical body of inner radius R . We assume a surrounding glycolipid layer of
thickness Lg , so the outer radius of the cell amounts to R  Lg . Note that our general
N2-fixing cell model accounts for both the situation of a heterocyst  Lg  0  and a
diazocyte  Lg  0  . The distribution of oxygen is described in two different zones: the
glycolipid layer (zone 1) and the diffusive boundary layer round the N2-fixing cell
(zone 2). To each zone we can apply the mass conservation equation
C 1   2 C 

r D

t r 2 r 
r 
[1]
where C denotes the oxygen concentration and  the consumption rate of oxygen at
a particular point. To simplify [1] no oxygen consumption is thought to occur in the
glycolipid layer. Similarly, no oxygen is consumed in the diffuse boundary layer
around the heterocyst. Furthermore, the overall mass exchange process is assumed to
reside in a steady-state. With these assumptions, equation [1] reduces to
1   2 C 
r D
0
r 2 r 
r 
[2]
in both the zones 1 and 2. The general solution to this second-order homogeneous
differential equation is given by (Boudreau, 1997)
C r   
A
B
r
[3]
where A and B denote integration constants that need to be determined via an
appropriate set of boundary conditions. Equally, the associated oxygen flux at a
particular point is given by
J  r   4 r 2 D
C
 4 DA
r
[4]
Boundary conditions
1
Far from the N2-fixing cell, the oxygen concentration approaches the bulk
concentration of the water. We assume a boundary layer of finite thickness Lb , i.e. the
oxygen concentration approaches the bulk concentration C at some distance
R  Lg  Lb
C r R L
g  Lb
 C
[5]
Inside the N2-fixing cell, oxygen consumption by respiration is assumed sufficiently
high to maintain a zero concentration value at the inner surface R of the glycolipid
layer
C r R  0
[6]
Finally, at the interface between the glycolipid layer and the surrounding water, i.e. r
= R  Lg , the continuity of both concentration and flux are stated
lim C  lim C
r  R  Lg 


lim J  lim J
r  R  Lg  
r  R  Lg 
[7]
r  R  Lg 
[8]
Model solution
Implementation of the boundary conditions [5]-[8] and the model equations [3]-[4]
leads to following set of equations
A1
 B1  0
Ri
A
 2  B2  C
RB
A
A
 1  B1   2  B2
Ro
Ro
[10]
D1 A1  D2 A2
[12]

[9]
[11]
where the subscripts 1 and 2 designate the particular zone in which the coefficients A
and B are valid. Furthermore, the diffusion efficiency  of the glycolipid layer is
introduced as a scaling factor relative to the diffusion coefficient in the surrounding
water i.e. D1   D2 . Upon combination of these equations, one eventually obtains

1
 1
A1  C  

 R R  Lg

 
1
1

   
  R  Lg  Lb R  Lg
2


 


1
[13]
Upon substitution of [13] into [4], one obtains that the total flux J through an
arbitrary shell around the heterocyst attains the constant value

1
1  1
J  4 DC   

   R R  Lg

 
1
1

  
  R  Lg  Lb R  Lg


 


1
[14]
Expression [14] represents the O2 flux through the cell membrane as predicted by our
general reactive-transport model of a N2-fixing cell. Two end-member situations can
be distinguished. In the case of a heterocyst, a thick impermeable glycolipid layer acts
as a diffusion barrier for oxygen. Hence, the thickness of the glycolipid layer is
considerable (i.e. Lg  0 ), and we can assume that the diffusion coefficient inside the
glycolipid layer is significantly lower than in the surrounding boundary , i.e. D1<<D2
or equally ε<<1. Then expression [14] readily reduces to

R
 
J  4 DC  R 1 

  Lg



 


[15]
A diazocyte forms a second end-member simplification model of expression [14]. A
diazocyte does not possess a glycolipid layer in contrast to a heterocyst, and hence, it
forms no diffusive barrier (i.e. zone 1 is completely absent). Accordingly, one can
substitute the values Lg  0 and   1 in expression [14], to finally obtain
 
R  
J  4 DC  R 1   
  Lb  
[16]
Reference list
1. Boudreau B.P. (1997) Diagenetic Models and Their Implementation. Springer, Berlin.
3
Rb  R  Lg  Lb
R
Rg  R  Lg
C
r
J
R
Rg
Rb
r
Figure A1. The oxygen concentration  C  and the oxygen flux  J  as a
function of the distance r from the centre, as predicted by the reactive
transport model of a general of the N2 fixing cell (assuming as spherical
geometry).
4