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Akberdin_etal

VIEWS: 7 PAGES: 17

									      Mathematical model of the
Arabidopsis thaliana L. morphogenesis
    in a cellular automaton terms

  I. R. Akberdin*1, E. A. Ozonov2, V.V. Mironova1, D.N. Gorpinchenko1,
      N. A. Omelyanchuk1, V. A. Likhoshvai1,2, N. A. Kolchanov1,2

   1-Institute of Cytology and Genetics SB RAS, Novosibirsk, Russia
                    2-Novosibirsk State University
                    e-mail: akberdin@bionet.nsc.ru
                        *Corresponding author

                http://www.bionet.nsc.ru/labs/modelgroup/
Integration of the experimental fragments:
Subject of the research: Arabidopsis thaliana L.
                      Advantages of Arabidopsis
                                                           Arabidopsis was first plant the
                        as model organisms::                      The Basic
                                                        genome of that was sequenced in full.
                     1)   Least genome from
                                                                   phytohormones:
                                                        There is research programe in that it’s
                                                          planned to define functions of all
                          khowns in higher plants
                          (125 mn. base pairs)                1. genes by – basic plant hormone
                                                                   Auxin 2010 years.
                                                                  regulating cells division and it’s
                     2)   Only 5 chromosomes                      the differentiation factor - Indole-
                                                         Number of papers about expression
                                                                  3-acetic     acids      (IAA      or
                                                          of the known genes goes up each
                                                                  heteroauxin).
                     3)   Short life cycle
Shoot apical meristem                                                           year
 (SAM) with forming        (~35 days)
        leaves
                                                                      2. Cytokinin     -   plant hormone
                     1)   A large number of seeds                        originating from 6-aminopurine;
                                                                         The basic natural cytokinin is
                          (to 5000 seeds in one                          zeatin (his synthetic analog is
                          plant)                                         kinetin кинетин);     Cytokinin in
                                                                         charges       of       totipotency
                     2)   Arabidopsis is un                   H   H      maintenance.
                                                              C   C
                          pretentious and easy                HC  H
                                                         C
     Arabidopsis          cultivated in lab
                                                    H
                                                    N    H2
                                                                  C
                                                              C N H
      thaliana            conditions including in
                                                N                 N
                                                                           6-phurphurilaminopurine
                          vitro
                                                                  N
                                                    N             H                        Vienna, YSF, 2007
                                    Objective:
Aim of the study:

Analysis of mechanisms that regulate maintenance of pluripotency
and cell differentiation during shoot apical meristem development in
A. thaliana
                                          2
                                   S               P    P
Tasks:                       kf  
                                  K 
                                           kr  1  2
                                   m,S          K m,P1 K m,P2
         V  e0              2
                  S  S            P         P  P             P      P
                  of         1  1  1   2 biosynthesis;
                     gene network of auxin  1  2
Reconstruction K  K  K  K  K
                   S   m,S        m,P1       i,P 
                                                   1     m,P2   K m,P1 K m,P2

Development of mathematical model that describes auxin metabolism
with account of genetic regulation;

Development of spatial distributed model that describes development
of the SAM in embryogenesis with account of genetic regulation.



                                                                                Vienna, YSF, 2007
                             Auxin biosynthesis gene network
                              is reconstructed by GeneNet*:
                                                                                                    Type objects                 Total objects
                                                                                                    compartment 8
                                                                                                    process                      7
                                                                                                    protein                      44
                                                                                                    reaction                     228
                                                                                                    gene                         62
                                                                                                    rna                          56
                                                                                                    cell                         40
                                                                                                    liter                        105
                                                                                                    species                      1
                                                                                                    repressor                    6
                                                                                                    substance                    1
* - Ananko E.A., Podkolodny N.L., Stepanenko I.L., Podkolodnaya O.A., Rasskazov D.A., Miginsky D.S., Likhoshvai V.A., Ratushny
A.V., Podkolodnaya N.N., Kolchanov N.A. (2005) GeneNet in 2005. Nucleic Acids Res., 33, D425-D427                                Vienna, YSF, 2007
                                    Cellular automaton model:
                                                                The Basic principles:
                                                                    The cells in the model may receive and, depending
                                                                    Type – Cell type; produce signals that neither
                                                                     on the cell type,
                                                                 Null. These cells mean empty spacethat should be
                                                                    BS0, SS0, bynor meaningful in the the
                                                                     accepted                  cells
                                                                 produce signalsSDother the values of model. Three
                                                                                          divide.
                                                                     biologically      0 –                   signals,           which
                                                                    signals Cells of the suspensor. They produce BS
                                                                 NullSus. producedstimulate the morphodynamics of
                                                                     unambiguously by the cell;
                                                                 (Basal Signal) and are confined to accepted level of
                                                                    BS, SS, SDThere the generally the lower part of
                                                                     cell tissues -at the are two the signals
                                                                                            values
                                                                 the abstraction, were selected ofNullSus cells in the
                                                                        embryo.                       :
                                                                    accepted by the cell;
                                                                 model.        SS is Stem Signal (most likely it is cytokinin);
                                                                    Kij – These cells imitate “differentiated” cells,
                                                                                characteristic Signal (most likely it
                                                                 Lateral.the SD is Differentiation of cell state at is auxin);
                                                                               BS SD (Signal (most the it is analogy
                                                                 which produce j) Basal Signalof Differentiation). of the
                                                                    position (i, is calculated as likely ratio of
                                                                 Promeristem. );
                                                                               auxin Cells of the embryo meristems.
                                                                    SS to SD. At the current point of time,
                                                                     All         produce in
                                                                These the embryo cells SS the model can be and are        classified
                                                                                                     in theSignal)
                                                                    the cells the the typepart(Stemthe embryo. During
                                                                                                                   state
                                                                          cell is influenced the signal they produce;
                                                                     according
                                                                 confined to to upper of of
                                                                    characterized by the parameter;L2meristem and
                                                                 development, they change into
                                                                    TCell the type cells. cell division, which
                                                                           types and of
                                                                L3meristem periodduration of the cell cycle depend on
                                                                      ij–
                                                                     the local signal concentration; They also produce
                                                                 Transit. Cells near the meristem.
                                                                    depends on the current value of the
                                                                    characteristicthe cell rates ofdepends
                                                                 SD, but have the K ; highest             division.
                                                                     Direction of
                                                                L2meristem. Cells ofdivisionmeristem. on gradients
                                                                                       ij
                                                                                                the                     They are
  Directions of the model’s                                          of –inthe number of
                                                                    Tpijthe signal distribution.iterations after
                                                                 situated      second layer from the epidermic layer of
                                                                              division of the cell These cells
                                                                    the last part of the embryo. at position produce
                                                                 the upper
      hypothetical signal                                        SS. j).
                                                                    (i,
 distributions in heart-stage                                    L3meristem. Cells located one layer down from the
        embryo tissues                                           L2meristem type cells. These cells produce SS.
Berleth T. and Chatfield S. “Embryogenesis: Pattern Formation from a Single Cell”. The Arabidopsis Book,
Published Online: September 30, 2002. http://www.bioone.org/perlserv/?request=get-abstract&doi=10.1199%2Ftab.0051   Vienna, YSF, 2007
                       Structure of the stem meristem cell niche:
Arbitrary units of cell division
           periods




                                              Ratio of auxin to cytokinin


                                                                    Vienna, YSF, 2007
              Interactions between model “cells” :

                                             The system of ordinary 
                                                                       n

                                                    
                                       BS ij   ij BS 0km e R
                                                   B
                                      differential equations is Bcalculated
                                                     for each cell:
                                       d
                                            SSi  SSi0  D  ( SS j  RS i )   SSi
                                                                        n
                                                                      SS
                                       
                                                    
                                        SSij   ij SS 0km e
                                       d
                                         dt        S
                                                           j

                                       
                                            SDi  SDi0  D  ( SD j SDi )   SDi
                                                                        n
                                       
                                       SDij   ij
                                       d
                                         dt         D
                                                         SD0 km e RD
                                                             j



 “Globalinteraction”:
 “Local interaction”:                  BSi  BSi  D  ( BS j  BSi )   BSi
                                                      0

                                        dt
                                                            j
    n=|i - boundary m|, whereare jequal to considered cell coordinates,
                     |j - conditions i, are
  Initial andk| + affected cell coordinates;thezero for all
    k,m are the
signals , R , R are the constants, which characterize penetrance for BS, SS
    ER ,RB    S    D


    and SD, respectively
  D- diffusion constant,
    – degradation the constants, signal. These constants express the
                 are constant,
  α cell to a certain type of which characterize the sensitivity ofcells
     , , ,
     E
    ij
          B
         ij
               S
              ij
                    D
                   ij



    receptivity to constants of a constitutive synthesis
  SS0, SD0, BS0 –signal.
depending on cell types (0, or >0).
                                                               Vienna, YSF, 2007
                Results of model.
            Normal development of SAM:
                                                                             - Meristem cells




                                                                             -L1 cells
                                              -Differentiation
 I              II                      III   cells

                     -Suspensor cells

                                                                 IV
I - 16 cell embryo (       - L1 cells,   - Promeristem,      - Lateral,
- Suspensor); II-globular stage (not new cell types), III-heart stage (
- Transit), IV- torpedo stage (   -L2meristem,    - L3meristem);

                                                                  Vienna, YSF, 2007
                                    Results of model.
                               Mutant development of SAM :
       Mutant II type:
       Mutation Itype:
       Model: The threshold value of parameter K to Promeristem cell is loweris higher in the mutant wild type
                   The sensitivity of Promeristem cell in Signal of Differentiation in mutants than in the than in the
       wild type; The sensitivity of L2, L3meristem cells to Stem Signal is lower in the mutant than in the wild type.
       Organism: wus-1 mutant
       Organism: cuc1 cuc2 mutant




           II               II               II III                           III                                          IV
                                                                                                               IV

        I - 16 cell embryo (       - L1 cells,   - Promeristem,      - Lateral,
        - Suspensor); II-globular stage (not new cell types), III-heart stage (
        - Transit), IV- torpedo stage (   -L2meristem,    - L3meristem);

 Aida M., Ishida T., Tasaka M. (1999). Shoot apical meristem and cotyledon formation during Arabidopsis embryogenesis:     Vienna,     YSF, 2007
 Interaction among the CUP-SHAPED COTYLEDON and SHOOT MERISTEMLESS genes. Development. 119. 823–831
Sharma V.K. and Fletcher J.C. (2003). Maintenance of Shoot and Floral Meristem Cell Proliferation and Fate. PNAS. 100. 11823- 11829.
                    Analyze sensitivity model
                  to SD (differentiation signal)
                   penetrance and synthesis :
     350
              400
                Normal
     300      350

     250      300
                                                             Mutation II type
              250
     200
  Num ber
Number        200
                                                                                                                    Mutation I type
of of cells
   cells
     150      150

     100      100
                  50
       50
                    0
        0                 1         3           5     7           9     11 13 15 17 19 21 23 25 27 29
            0.1
                  0.3
                        0.5
                              0.7
                                    0.9
                                          1.1
                                                1.3
                                                      1.5
                                                            1.7
                                                                  1.9
                                                                        2.1
                                                                              2.3
                                                                                    2.5
                                                                                          2.7
                                                                                                2.9
                                                                                                      3.1
                                                                                                            3.3
                                                                                                                  3.5
                                                                                                                        3.7
                                                                                                                              3.9
                                                                                                                                    4.1
                                                                                                                                          4.3
                                                                                                                                                4.5
                                                                                                                                                      4.7
                                                                                                                                                            4.9
In developing SAM                                             differentiation signal synthesis*
                                                          differentiation signal penentrance*

Influence of the SD (differentiation signal) penetrance on cellular automaton
Influence of the SD (differentiation signal) synthesis on the the cellular automaton evolution                                              Vienna, YSF, 2007
(* - parameter values are in arbitrary arbitrary
evolution (* - parameter values are inunits). units)
           Visualization of the cellular
               automaton model:




                         Stem signal




                          Basal signal


Differentiation signal

                                           Vienna, YSF, 2007
                 Conclusions:
              Future prospects:
 consider in the model more:
 complex geometry of cell interactions,
The GeneNet technology was used to reconstruct the auxin
biosynthesis gene network;
 molecular     processes    regulating
We constructed a mathematical active             transport
                                         model of the SAM  of
 morphogenes and
development in embryogenesis of Arabodopsis thaliana;
 The model adequately describes two processes: cell
 differentiation and maintenance of their cell abnormal
                                      stem biosynthesis and
In this cellular automaton we modeled population;SAM
 genetic processes regulating
conjugation-degradation!!!
development in two mutants: cuc1 cuc2, wus-1;
Early development step of the SAM was adequately
simulated without    considering active transport of Basic
signal. Embryo anisotropy is formed mainly by Basal signal.
Processes of the signal distribution can be described in
diffusion framework; is more affected by parameters
SAM development
responsible for morphogene transport than by parameters
related to morphogene biosynthesis.

                                                 Vienna, YSF, 2007
             Acknowledgements:


Mathematical and programmer support:
Phd Likhoshvai V.A.
BSc Ozonov E.A.
BSc Gorpinchenko D.N.

Informational and biological data support:
Senior scientist Omelyanchuk N.A.
Phd student, Mironova V.V.

Dr., Prof. Kolchanov N.A.

                                 Vienna, YSF, 2007
Thank you very much!!!
      Methods of regulator pathway
   reconstruction of the gene networks:

а)Correlation methods - алгоритм основан на условной корреляции
    равновесных значений концентраций мРНК двух генов, один из
    которых нокаутирован
(J. Rice, Y. Tu and G. Stolovitzky, Bioinformatics, 2005)
б)    Examine      –    алгоритм    использует    линейную  модель
    дифференциальных уравнений, и адаптивный оптимизационный
    алгоритм для поиска модели максимально удовлетворяющей
    экспериментальным данным, путем повышения степеней захода
    вершин графа и перехода к новой стадии перебора, пока не
    найдется наилучшая модель
(X. Deng , H. Geng ,H. Ali, Biosystems, 2005)
в) Boolean gene network (Reveal) - алгоритм рассчитывает
    взаимную      информацию       и     энтропию     между  двумя
    последовательными состояниями, а также анализирует все
    возможные булевы функции от малого количества переменных, и
    по их отношению определяет множество предшественников для
    вершины графа.
(S. Liang , S. Fuhrman, R. Somogyi, Pac. Symp. Biocomputers,
    1998)
                                                     Томск, 2007
      MGSmodeller – computer system for
reconstruction, calculation and analyze models o
         the molecular-genetic systems:




                                     Томск, 2007

								
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