A plausible mechanism for auxin patterning along the developing root

<p>Abstract</p> <p>Background</p> <p>In plant roots, auxin is critical for patterning and morphogenesis. It regulates cell elongation and division, the development and maintenance of root apical meristems, and other processes. In <it>Arabidopsis</it>, auxin distribution along the central root axis has several maxima: in the root tip, in the basal meristem and at the shoot/root junction. The distal maximum in the root tip maintains the stem cell niche. Proximal maxima may trigger lateral or adventitious root initiation.</p> <p>Results</p> <p>We propose a <it>reflected flow </it>mechanism for the formation of the auxin maximum in the root apical meristem. The mechanism is based on auxin's known activation and inhibition of expressed PIN family auxin carriers at low and high auxin levels, respectively. Simulations showed that these regulatory interactions are sufficient for self-organization of the auxin distribution pattern along the central root axis under varying conditions. The mathematical model was extended with rules for discontinuous cell dynamics so that cell divisions were also governed by auxin, and by another morphogen <it>Division Factor </it>which combines the actions of cytokinin and ethylene on cell division in the root. The positional information specified by the gradients of these two morphogens is able to explain root patterning along the central root axis.</p> <p>Conclusion</p> <p>We present here a plausible mechanism for auxin patterning along the developing root, that may provide for self-organization of the distal auxin maximum when the <it>reverse fountain </it>has not yet been formed or has been disrupted. In addition, the proximal maxima are formed under the <it>reflected flow </it>mechanism in response to periods of increasing auxin flow from the growing shoot. These events may predetermine lateral root initiation in a rhyzotactic pattern. Another outcome of the <it>reflected flow </it>mechanism - the predominance of lateral or adventitious roots in different plant species - may be based on the different efficiencies with which auxin inhibits its own transport in different species, thereby distinguishing two main types of plant root architecture: taproot vs. fibrous.</p

Pluripotency gene network dynamics: System views from parametric analysis.

Multiple experimental data demonstrated that the core gene network orchestrating self-renewal and differentiation of mouse embryonic stem cells involves activity of Oct4, Sox2 and Nanog genes by means of a number of positive feedback loops among them. However, recent studies indicated that the architecture of the core gene network should also incorporate negative Nanog autoregulation and might not include positive feedbacks from Nanog to Oct4 and Sox2. Thorough parametric analysis of the mathematical model based on this revisited core regulatory circuit identified that there are substantial changes in model dynamics occurred depending on the strength of Oct4 and Sox2 activation and molecular complexity of Nanog autorepression. The analysis showed the existence of four dynamical domains with different numbers of stable and unstable steady states. We hypothesize that these domains can constitute the checkpoints in a developmental progression from naïve to primed pluripotency and vice versa. During this transition, parametric conditions exist, which generate an oscillatory behavior of the system explaining heterogeneity in expression of pluripotent and differentiation factors in serum ESC cultures. Eventually, simulations showed that addition of positive feedbacks from Nanog to Oct4 and Sox2 leads mainly to increase of the parametric space for the naïve ESC state, in which pluripotency factors are strongly expressed while differentiation ones are repressed

Pluripotency gene network dynamics: System views from parametric analysis - Fig 5

<p><b>a-b:</b> Time series of mRNA and protein expressions for <i>Nanog</i> at <i>h = 10</i>: <i>v</i>_<i>1</i>—<i>Nanog</i> mRNA concentration, <i>v</i>_<i>2</i> –Nanog protein concentration in the nucleus, <i>v</i>_<i>3</i>—Nanog protein concentration in the cytoplasm; The insets in Fig 5A and 5B represent the same curves as on the main part, but with a zoomed scale of the y-axis. <b>c-d:</b> Time series for concentrations of pluripotent (<i>w</i>_<i>1</i>) and differentiation (<i>w</i>_<i>2</i>) factors. Concentration oscillations of Nanog and pluripotent/differentiation factors occurred at <i>A = 0</i>.<i>2</i> (a, c) and <i>A = 0</i>.<i>3</i> (b, d). The other parameters were fixed. c: The pluripotent factors <i>w</i>_<i>1</i> were suppressed and the differentiation factors <i>w</i>_<i>2</i> were expressed. This state corresponds to differentiation. d: Pluripotent factors were highly expressed, and differentiation factors were suppressed. This state corresponds to pluripotency.</p

Pluripotency gene network dynamics: System views from parametric analysis - Fig 1

<p><b>A:</b> The core transcriptional network of the factors orchestrating the pluripotency and differentiation genes (suggested by [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194464#pone.0194464.ref010" target="_blank">10</a>]). External A₊ and B_- signals activate and repress expression of <i>Oct4</i>, <i>Sox2</i> and <i>Nanog</i> genes, correspondingly. Oct4 and Sox2 form a heterodimer, Oct4/Sox2, which positively regulates <i>Oct4</i>, <i>Sox2</i> and <i>Nanog</i> expression. Nanog directly induces <i>Oct4</i>, <i>Sox2</i> and its own expression. Oct4/Sox2 heterodimer and Nanog positively regulate pluripotency genes and repress differentiation genes. <b>B:</b> The revised core gene network suggested in this paper, in which transcription and translation processes were added; external signal B- is removed and positive signal A+ activates transcription of <i>Oct4</i> и <i>Sox2</i> genes. Nanog represses its own transcription and does not influence on <i>Oct4</i> and <i>Sox2</i> expression.</p

Multiplicity of stationary solutions representing as <i>w</i>_<i>1</i> and <i>w</i>_<i>2</i> dependence on the parameter <i>A</i>, <i>0</i> ≤ <i>A</i> ≤ <i>0</i>.<i>4</i> at <i>h = 6</i> (2a and 2b) and <i>h = 2</i> (2c and 2d).

<p>a. Initial steady state values <i>w</i>_<i>2</i>><i>w</i>_<i>1</i> and <i>A = 0</i> simulate differentiation state. The <i>w</i>_<i>1</i><i>/w</i>_<i>2</i> ratio while A is growing upon <i>A</i> = <i>A</i>_<i>*</i> = <i>0</i>.<i>277</i> (the turning point), corresponds to the differentiation steady state. Asterisks indicate arcs of the graphs with unstable solutions. b. Initial steady state values <i>w</i>_<i>1</i>> <i>w</i>_<i>2</i> and <i>A</i> ˃ <i>A</i>_<i>*</i> simulate the pluripotent state. The graph of the steady state while decreasing A upon <i>A ≥ 0</i> (including the range <i>0</i> ≤ <i>A</i> ˂ <i>A</i>_<i>*</i>) is depicted. Both Fig 2A and 2B show that three states (two stable and one unstable) exist in the range 0 ≤ <i>A</i> ˂ <i>A</i>_<i>*</i> <i>= 0</i>.<i>277</i>, while there is one steady state, when <i>A</i> ˃ <i>A</i>_<i>*</i>. c. Stationary solution with initial steady state values <i>w</i>_<i>2</i>><i>w</i>_<i>1</i> and A = 0 corresponding to the differentiated cell. <i>w</i>_<i>1</i> and <i>w</i>_<i>2</i> variables have turning points at <i>A</i> = <i>A</i>_<i>*</i>. Asterisks indicate unstable solutions. d. Stationary solution with initial steady state values corresponding to the pluripotent cell. The Fig 2C and 2D shows that three states (two are stables and one is unstable) exist in the range <i>0</i> ≤ <i>A</i> ˂ <i>A</i>_<i>*</i> <i>= 0</i>.<i>277</i>, while there is one steady state, when <i>A</i> ˃ <i>A</i>_<i>*</i> and this is the pluripotent state only.</p

The bistable switch in the core network depending on (<i>a</i>_<i>3</i>, <i>a</i>_<i>7</i>) parameters and at <i>h = 6</i>. Highlighted region is the parameters range, for which the switch-like behavior has existed.

<p>Furthermore, the analysis indicated that a straight line <i>a</i>_<i>3</i> = <i>a</i>_<i>7</i> in the plane (<i>a</i>_<i>3</i>, <i>a</i>_<i>7</i>) divides it into two areas. When <i>a</i>_<i>3</i> < <i>a</i>_<i>7</i>, there will be some <i>A</i>_<i>0</i>, that upon <i>A > A</i>_<i>0</i> the cell is pluripotent, while at <i>A < A</i>_<i>0</i> the cell will differentiate. When <i>a</i>_<i>3</i> > <i>a</i>_<i>7</i>, the cell has pluripotent state at all values <i>A</i> ≥ <i>0</i>.</p

Biological relevance and values of the model parameters.

<p>Biological relevance and values of the model parameters.</p

The bistable switch in the core network depending on (<i>a</i>_<i>3</i>, <i>a</i>_<i>7</i>) parameters and at <i>h = 6</i>.

<p>Highlighted region is the range of parameter values, having which the system has switch-like behavior. Furthermore, the analysis indicated that a straight line <i>a</i>_<i>3</i> = <i>a</i>_<i>7</i> divides the plane (<i>a</i>_<i>3</i>, <i>a</i>_<i>7</i>) it into two areas. When <i>a</i>_<i>3</i> < <i>a</i>_<i>7</i>, the cell has differentiated state at all values <i>A</i> ≥ <i>0</i>. When <i>a</i>_<i>3</i> > <i>a</i>_<i>7</i>, there will be some <i>A</i>_<i>0</i>, that upon <i>A</i> > <i>A</i>_<i>0</i> the cell is pluripotent, while at <i>A</i> < <i>A</i>_<i>0</i> the cell will differentiate.</p

Multiplicity and stability of stationary solutions depending on parameters <i>A</i> and <i>h</i>. <i>D</i>_<i>4</i> domain comprises a single stable steady state, pluripotency; <i>D</i>_<i>2</i>domain encompasses a single unstable state (oscillation).

<p><i>D</i>_<i>3</i> domain includes three unstable states (oscillations); <i>D</i>_<i>4</i> domain contains three states, from which two (pluripotency and differentiation) are stable and one (transition between these states is unstable (according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194464#pone.0194464.g002" target="_blank">Fig 2A</a>). Domains (a) predicted by the model and (b) their correspondence to developmental progression of ESCs from the naïve pluripotency (the ground state) to lineage commitment according to [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194464#pone.0194464.ref061" target="_blank">61</a>]. The initial phase of exit from the ground state is asynchronous in the cell population and reversible until the complete dissipation of naïve state factors (reviewed in [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194464#pone.0194464.ref059" target="_blank">59</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194464#pone.0194464.ref061" target="_blank">61</a>, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0194464#pone.0194464.ref062" target="_blank">62</a>]). Cells reaching a transitional point after 2i withdrawal are competent for lineage specification and characterized by the absence of both groups, naïve factors and lineage markers. The late phase of pluripotency (primed pluripotency) is characterized by nascent expression of lineage specification factors. The “clock model” was proposed as a route of consistent transitions with the dual mechanism of hour hand movement depending on the initial cell state: pluripotent (counter-clockwise movement of black solid arrows) or differentiated (clockwise movement of black solid arrows). Red arrows, in turn, reflects directions from naïve to reverse-transition-primed stages (developmental progression during differentiation) or from primed to transition-reverse-naïve states (developmental progression during reprogramming into pluripotent state), while dotted black arrows were added to underline intermediate reverse and transitional states to which domains <i>D</i>_<i>2</i> and <i>D</i>_<i>3</i> correspond, respectively.</p