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Supplementary information
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Conservation of meristem number
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Let consider a generalized volume  defining both the physical domain (r =
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(x,y) in 2D) and the domains defining the directions of the meristems (u =
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(cosα,sinα) in 2D). The number of meristems (denoted ρ to simplify the
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notations) contained in this volume is therefore determined as   .dvdu . The

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number of meristems in this volume is changing as a result of meristems
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moving in and out of the volume by expansion e but also due to reorientation of
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meristems due to gravitropism g and the creation of new meristems due to
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branching b. These fluxes of meristems at the surface  of the domain  are
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determined by the surface integral
 V .ndS
where V = (eu,g) is the

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generalized velocity of meristems and n is the normal vector along the surface
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.
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The principle of conservation of the number of meristems implies:
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
.dvdu   V .ndS   b.dvdu
t 


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Using Gauss formula, this equation then becomes
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
 t

   ( V )dvdu   b.dvdu

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1
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This equation is valid for every volume , and this imply that the differential
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form of the balance law can be written as:

  *  ( g )    ( eu )  b
t
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Where * denote operators to coordinates related to direction of growth u, and b
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denotes the increase of meristem density due to branching.
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Analytical solution of 1D meristem dynamics using the method of
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characteristics
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The system in equation 4 can be solved by finding the solution for the dynamics
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of the roots of the first order. For the sake of simplicity, we will assume that
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root angle is defined from the vertical direction. In one spatial dimension, the
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problem is therefore defined as:
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 t  (g )  e cos  z  0
Eqn 9
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We denote X(t) a trajectory of a root in the z,α space:
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z (t ) 
~
X (t )   ~ 
  (t ) 
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The characteristics of the equation is defined by:
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 e cos ~ 
X ' (t )  
~ 
  g 
Eqn 10
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This equation can be solved by finding the solution ~ (t ) using a 2nd order taylor
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polynomial approximation for the cosine in equation 10:
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2
~

e~
 z 0  et  0 (e  2 gt  1) 
X (t )  

4g
~ e  gt



0


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Along this characteristic curve, Eqn 9 simplify to a second ordinary differential
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equation (Eqn 11):
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d ( X (t ), t )
  g ( X (t ), t )
dt
Eqn 11
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Solution of Eqn 11 using solution of Eqn 10 gives:
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 ( X (t ), t )   ( X 0 ,0) exp  gt 
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In order to obtain the solution of Eqn 9 the following change of variable is
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made:
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e~
z~
z 0  et  0 (e 2 gt  1)
4g
 gt
~
  e
0
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Replacing the expression of z0 and α0 in the solution of Eqn 11 gives the
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solution for Eqn 9:
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

e 2
 ( z,  , t )    z  et 
(1  e 2 gt ), e gt  exp  gt 
4g


0
Eqn 12
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It is convenient also to obtain the evolution of the profile of the total root
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meristem density. This is determined as the integral of ρ with regards to the
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angle α:
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 ( z,, t ) 
 /2
Eqn 13
  ( z, , t )d
 / 2
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We use the following change of variable α’ = αegt to simplify this integral:
 ( z,, t ) 
 exp( gt ) / 2
  z  et  e
0
2

/ 4 g (e 2 gt  1),  d
 exp( gt ) / 2
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We can further simplify this integral by assuming the angle and depth of initial
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root distribution is not correlated. Therefore, the initial root meristem
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distribution is the product of two functions such that:
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 0 ( z, )  10 ( z )( )
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Where П is the rectangle function such that it value is 1 between 0 and π.
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Therefore, the dynamics of first order root meristem density is found to be:
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1 ( z,, t ) 
 exp( gt ) / 2
  z  et  e
0
2

/ 4 g (e 2 gt  1),  d
 exp( gt ) / 2
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The dynamics of second order lateral roots differs from the first order only by
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the source term and initial conditions. If we assume for simplification that
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branching is proportional to ρ1 and that both type of roots have the same
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behaviour, the total root meristem density will then be written as:
 ( z,, t )  (1  bt )
 exp( gt ) / 2
0
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 2 gt
  z  et  e / 4 g (e  1),  d
 exp( gt ) / 2
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9
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Eqn 14