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Floral morphogenesis: stochastic explorations of a gene network epigenetic landscape.

Alvarez-Buylla ER, Chaos A, Aldana M, Benítez M, Cortes-Poza Y, Espinosa-Soto C, Hartasánchez DA, Lotto RB, Malkin D, Escalera Santos GJ, Padilla-Longoria P - PLoS ONE (2008)

Bottom Line: Thus, time ordering in the emergence of cell-fate patterns is not an artifact of synchronous updating in the Boolean model.Therefore, our model provides a novel explanation for the emergence and robustness of the ubiquitous temporal pattern of floral organ specification.It also constitutes a new approach to understanding morphogenesis, providing predictions on the population dynamics of cells with different genetic configurations during development.

View Article: PubMed Central - PubMed

Affiliation: Instituto de Ecología, Universidad Nacional Autónoma de México, Cd. Universitaria, México, D. F., México. elena.alvarezbuylla@gmail.com

ABSTRACT
In contrast to the classical view of development as a preprogrammed and deterministic process, recent studies have demonstrated that stochastic perturbations of highly non-linear systems may underlie the emergence and stability of biological patterns. Herein, we address the question of whether noise contributes to the generation of the stereotypical temporal pattern in gene expression during flower development. We modeled the regulatory network of organ identity genes in the Arabidopsis thaliana flower as a stochastic system. This network has previously been shown to converge to ten fixed-point attractors, each with gene expression arrays that characterize inflorescence cells and primordial cells of sepals, petals, stamens, and carpels. The network used is binary, and the logical rules that govern its dynamics are grounded in experimental evidence. We introduced different levels of uncertainty in the updating rules of the network. Interestingly, for a level of noise of around 0.5-10%, the system exhibited a sequence of transitions among attractors that mimics the sequence of gene activation configurations observed in real flowers. We also implemented the gene regulatory network as a continuous system using the Glass model of differential equations, that can be considered as a first approximation of kinetic-reaction equations, but which are not necessarily equivalent to the Boolean model. Interestingly, the Glass dynamics recover a temporal sequence of attractors, that is qualitatively similar, although not identical, to that obtained using the Boolean model. Thus, time ordering in the emergence of cell-fate patterns is not an artifact of synchronous updating in the Boolean model. Therefore, our model provides a novel explanation for the emergence and robustness of the ubiquitous temporal pattern of floral organ specification. It also constitutes a new approach to understanding morphogenesis, providing predictions on the population dynamics of cells with different genetic configurations during development.

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Temporal sequence of cell-fate attainment patterns under the Boolean dynamics with noise.Maximum relative probability (“Y” axis) of attaining each attractor, as a function of iteration number or time (“X” axis). (A) Probability of attaining each attractor (i.e., cell type) obtained by multiplying the Markov matrix M by a population vector  initialized at the sepal attractor. The error probability in computing this graph was η = 0.03. The most probable sequence of cell attainment is: Sepals, petals, carpels, and stamens. (B) Probability of attaining each attractor (i.e., cell type) at each iteration when 80000 randomly chosen “sepal” configurations were selected and followed for 140 steps. Noise was introduced in the updating of each gene independently, with a η = 0.03 probability at each iteration. The probabilities for the petal (p) and stamen (st) attractors correspond to the sum of p1+p2 and st1+st2, respectively. All maxima correspond to 100 because each absolute probability value was divided by the maximum of each attractor's curve (see Results and Methods). Equivalent graphs to those in (A) and (B) for η = 0.01 are shown in (C) and (D), respectively.
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pone-0003626-g003: Temporal sequence of cell-fate attainment patterns under the Boolean dynamics with noise.Maximum relative probability (“Y” axis) of attaining each attractor, as a function of iteration number or time (“X” axis). (A) Probability of attaining each attractor (i.e., cell type) obtained by multiplying the Markov matrix M by a population vector initialized at the sepal attractor. The error probability in computing this graph was η = 0.03. The most probable sequence of cell attainment is: Sepals, petals, carpels, and stamens. (B) Probability of attaining each attractor (i.e., cell type) at each iteration when 80000 randomly chosen “sepal” configurations were selected and followed for 140 steps. Noise was introduced in the updating of each gene independently, with a η = 0.03 probability at each iteration. The probabilities for the petal (p) and stamen (st) attractors correspond to the sum of p1+p2 and st1+st2, respectively. All maxima correspond to 100 because each absolute probability value was divided by the maximum of each attractor's curve (see Results and Methods). Equivalent graphs to those in (A) and (B) for η = 0.01 are shown in (C) and (D), respectively.

Mentions: Since we wanted to find the most probable sequence of transitions among the attractors representing the various cell types, we followed the changes in the probability of reaching a certain attractor throughout time given that the system was initialized in a particular attractor at time t = 0 (see Figure 3). In order to achieve this, note that the Markov matrix (herein denoted as M) in Table 1 contains the conditional probabilities P(n/m) of reaching attractor n at time t+τ, given that the system is at attractor m at time t. In order to obtain the temporal sequence in which attractors are most likely reached, it is necessary to repeatedly multiply the Markov matrix M by the vector , whose entries contain the fraction of cells at each attractor in a given population at time t. In other words, , where v1(t) is the fraction of cells in the population whose configurations at time t are in the basin of attraction of the first attractor, v2(t) is the fraction of cells at time t in the basin of attraction of the second attractor, and so on. Starting out from a population with a given distribution of cells among the attractors, the distribution of cells at time t is given by: .


Floral morphogenesis: stochastic explorations of a gene network epigenetic landscape.

Alvarez-Buylla ER, Chaos A, Aldana M, Benítez M, Cortes-Poza Y, Espinosa-Soto C, Hartasánchez DA, Lotto RB, Malkin D, Escalera Santos GJ, Padilla-Longoria P - PLoS ONE (2008)

Temporal sequence of cell-fate attainment patterns under the Boolean dynamics with noise.Maximum relative probability (“Y” axis) of attaining each attractor, as a function of iteration number or time (“X” axis). (A) Probability of attaining each attractor (i.e., cell type) obtained by multiplying the Markov matrix M by a population vector  initialized at the sepal attractor. The error probability in computing this graph was η = 0.03. The most probable sequence of cell attainment is: Sepals, petals, carpels, and stamens. (B) Probability of attaining each attractor (i.e., cell type) at each iteration when 80000 randomly chosen “sepal” configurations were selected and followed for 140 steps. Noise was introduced in the updating of each gene independently, with a η = 0.03 probability at each iteration. The probabilities for the petal (p) and stamen (st) attractors correspond to the sum of p1+p2 and st1+st2, respectively. All maxima correspond to 100 because each absolute probability value was divided by the maximum of each attractor's curve (see Results and Methods). Equivalent graphs to those in (A) and (B) for η = 0.01 are shown in (C) and (D), respectively.
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC2572848&req=5

pone-0003626-g003: Temporal sequence of cell-fate attainment patterns under the Boolean dynamics with noise.Maximum relative probability (“Y” axis) of attaining each attractor, as a function of iteration number or time (“X” axis). (A) Probability of attaining each attractor (i.e., cell type) obtained by multiplying the Markov matrix M by a population vector initialized at the sepal attractor. The error probability in computing this graph was η = 0.03. The most probable sequence of cell attainment is: Sepals, petals, carpels, and stamens. (B) Probability of attaining each attractor (i.e., cell type) at each iteration when 80000 randomly chosen “sepal” configurations were selected and followed for 140 steps. Noise was introduced in the updating of each gene independently, with a η = 0.03 probability at each iteration. The probabilities for the petal (p) and stamen (st) attractors correspond to the sum of p1+p2 and st1+st2, respectively. All maxima correspond to 100 because each absolute probability value was divided by the maximum of each attractor's curve (see Results and Methods). Equivalent graphs to those in (A) and (B) for η = 0.01 are shown in (C) and (D), respectively.
Mentions: Since we wanted to find the most probable sequence of transitions among the attractors representing the various cell types, we followed the changes in the probability of reaching a certain attractor throughout time given that the system was initialized in a particular attractor at time t = 0 (see Figure 3). In order to achieve this, note that the Markov matrix (herein denoted as M) in Table 1 contains the conditional probabilities P(n/m) of reaching attractor n at time t+τ, given that the system is at attractor m at time t. In order to obtain the temporal sequence in which attractors are most likely reached, it is necessary to repeatedly multiply the Markov matrix M by the vector , whose entries contain the fraction of cells at each attractor in a given population at time t. In other words, , where v1(t) is the fraction of cells in the population whose configurations at time t are in the basin of attraction of the first attractor, v2(t) is the fraction of cells at time t in the basin of attraction of the second attractor, and so on. Starting out from a population with a given distribution of cells among the attractors, the distribution of cells at time t is given by: .

Bottom Line: Thus, time ordering in the emergence of cell-fate patterns is not an artifact of synchronous updating in the Boolean model.Therefore, our model provides a novel explanation for the emergence and robustness of the ubiquitous temporal pattern of floral organ specification.It also constitutes a new approach to understanding morphogenesis, providing predictions on the population dynamics of cells with different genetic configurations during development.

View Article: PubMed Central - PubMed

Affiliation: Instituto de Ecología, Universidad Nacional Autónoma de México, Cd. Universitaria, México, D. F., México. elena.alvarezbuylla@gmail.com

ABSTRACT
In contrast to the classical view of development as a preprogrammed and deterministic process, recent studies have demonstrated that stochastic perturbations of highly non-linear systems may underlie the emergence and stability of biological patterns. Herein, we address the question of whether noise contributes to the generation of the stereotypical temporal pattern in gene expression during flower development. We modeled the regulatory network of organ identity genes in the Arabidopsis thaliana flower as a stochastic system. This network has previously been shown to converge to ten fixed-point attractors, each with gene expression arrays that characterize inflorescence cells and primordial cells of sepals, petals, stamens, and carpels. The network used is binary, and the logical rules that govern its dynamics are grounded in experimental evidence. We introduced different levels of uncertainty in the updating rules of the network. Interestingly, for a level of noise of around 0.5-10%, the system exhibited a sequence of transitions among attractors that mimics the sequence of gene activation configurations observed in real flowers. We also implemented the gene regulatory network as a continuous system using the Glass model of differential equations, that can be considered as a first approximation of kinetic-reaction equations, but which are not necessarily equivalent to the Boolean model. Interestingly, the Glass dynamics recover a temporal sequence of attractors, that is qualitatively similar, although not identical, to that obtained using the Boolean model. Thus, time ordering in the emergence of cell-fate patterns is not an artifact of synchronous updating in the Boolean model. Therefore, our model provides a novel explanation for the emergence and robustness of the ubiquitous temporal pattern of floral organ specification. It also constitutes a new approach to understanding morphogenesis, providing predictions on the population dynamics of cells with different genetic configurations during development.

Show MeSH