God's Will and the Origin of the World. ‘Abd al-Latif al-Baghdadi's Sources and Arguments

As well stated by Herbert Davidson in the Introduction to his famous book Proof for Eternity, Creation and the Existence of God in Medieval Islamic and Jewish Philosophy: \u201cThe issue of eternity and creation provided an arena for determining the relationship of God to the universe, for determining, specifically, whether God is a necessary or a voluntary cause\u201d,1 whether volition is a further attribute for deity in addition to the uncaused cause, the incorporeal and the One. This paper addresses \u2bfAbd al-La\u1e6d\u12bf al-Ba\u121d\u101d\u12b's position in this arena: his sources and arguments in the Book on the Science of Metaphysics (Kit\u101b f\u12b \u2018Ilm M\u101 ba\u2018d al-\u1e6dab\u12b\u2018a)

Dispersion relations for the spot patterns underlying the branching patterns in Figs 4 and 5.

<p>(A) Dispersion relations for the spot patterns underlying the branching patterns in Figs <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0174946#pone.0174946.g004" target="_blank">4</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0174946#pone.0174946.g005" target="_blank">5</a>. <i>λ</i> represents the eigenvalue with the largest real part for a given wavenumber <i>k</i>. <i>k</i>1-6 are the critical wavenumbers at which the maximum value of <i>λ</i> occurs at. <i>k</i>1-3 are the wavenumbers of the spot patterns in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0174946#pone.0174946.g004" target="_blank">Fig 4D–4F</a>, while <i>k</i>4-6 are the wavenumbers of the spot patterns in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0174946#pone.0174946.g005" target="_blank">Fig 5D–5F</a>. (B) Comparison of the wavelength (2π/<i>k</i>) of the spot patterns underlying the branching patterns in Figs <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0174946#pone.0174946.g004" target="_blank">4</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0174946#pone.0174946.g005" target="_blank">5</a>. Patterns 1–3 represent the spot patterns in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0174946#pone.0174946.g004" target="_blank">Fig 4D–4F</a>, while patterns 4–6 represent the spot patterns in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0174946#pone.0174946.g005" target="_blank">Fig 5D–5F</a>.</p

Effect of the wavelength on the branching patterns in the Turing region for <i>ρ</i>_<i>H</i> = 0.00013.

<p>(A) The cell differentiation trajectories of the branching patterns cross the Turing region. Points 1–6 correspond to the position of the underlying Turing patterns. (B) Branching patterns (top) and underlying spot patterns (bottom). Patterns 1–6 correspond to points 1–6 in A. (C) Dispersion relations for patterns, with wavenumber <i>k</i>1-6 corresponding to patterns 1–6 in B. (D) Comparison of the wavelength (1π/wavenumber) from patterns 1–6 in B. Parameters: <i>c</i> = 0.002, <i>μ</i> = 0.16, <i>ρ</i>_<i>A</i> = 0.03, <i>D</i>_<i>A</i> = 0.02, <i>v</i> = 0.04, <i>ρ</i>_<i>H</i> = 0.00013, <i>D</i>_<i>H</i> = 0.3, <i>c</i>₀ = 0.02, <i>D</i>_<i>S</i> = 0.06, (B1-6) <i>ε</i> = 1.1/0.85/0.7/0.07/0.06/0.045. In the 2D patterns, black indicates a high concentration, and gray indicates a low concentration.</p

Turing mechanism underlying a branching model for lung morphogenesis

<div><p>The mammalian lung develops through branching morphogenesis. Two primary forms of branching, which occur in order, in the lung have been identified: tip bifurcation and side branching. However, the mechanisms of lung branching morphogenesis remain to be explored. In our previous study, a biological mechanism was presented for lung branching pattern formation through a branching model. Here, we provide a mathematical mechanism underlying the branching patterns. By decoupling the branching model, we demonstrated the existence of Turing instability. We performed Turing instability analysis to reveal the mathematical mechanism of the branching patterns. Our simulation results show that the Turing patterns underlying the branching patterns are spot patterns that exhibit high local morphogen concentration. The high local morphogen concentration induces the growth of branching. Furthermore, we found that the sparse spot patterns underlie the tip bifurcation patterns, while the dense spot patterns underlies the side branching patterns. The dispersion relation analysis shows that the Turing wavelength affects the branching structure. As the wavelength decreases, the spot patterns change from sparse to dense, the rate of tip bifurcation decreases and side branching eventually occurs instead. In the process of transformation, there may exists hybrid branching that mixes tip bifurcation and side branching. Since experimental studies have reported that branching mode switching from side branching to tip bifurcation in the lung is under genetic control, our simulation results suggest that genes control the switch of the branching mode by regulating the Turing wavelength. Our results provide a novel insight into and understanding of the formation of branching patterns in the lung and other biological systems.</p></div

Dense spot patterns underlying different side branching patterns.

<p>(A-C) The side branching patterns. (D-F) The underlying spot patterns have a dense distribution that corresponds to the side branching patterns A-C. The spatial interval has a small decrease in the side branching patterns with the production of more branches as the number of spots slightly increases in the underlying spot patterns. Parameters: <i>c</i> = 0.002, <i>μ</i> = 0.16, <i>ρ</i>_<i>A</i> = 0.03, <i>D</i>_<i>A</i> = 0.02, <i>v</i> = 0.04, <i>ρ</i>_<i>H</i> = 0.0001, <i>D</i>_<i>H</i> = 0.3, <i>c</i>₀ = 0.02, <i>γ</i> = 0.02, <i>D</i>_<i>S</i> = 0.06, <i>d</i> = 0.008, <i>e</i> = 0.1, <i>f</i> = 10, (A-C) <i>ε</i> = 0.1/0.06/0.045, (D-F) <i>S</i> = 0.614, <i>Y</i> = 0.478; <i>S</i> = 0.679, <i>Y</i> = 0.510; <i>S</i> = 0.716, <i>Y</i> = 0.530. In the 2D patterns, black indicates a high concentration while gray indicates a low concentration.</p

Schematic diagram of the research scheme.

<p>Top: The branching model was decoupled to obtain the activator-inhibitor model. Bottom: The S-Y parameter space of the activator-inhibitor model was calculated for the Turing instability. A crescent-shaped Turing region is presented as a result. The SY-curve for the differentiation trajectory of a cell in the branching system was plotted in the S-Y plane. Each state (S, Y) of cell differentiation is represented as a point on the SY-curve. To acquire the Turing pattern underlying the branching pattern, a point (e.g., point p1) on the trajectory within the Turing region was taken as the values of Sand Y, and simulation of the activator-inhibitor model was performed.</p

Sparse spot patterns underlying different tip bifurcation patterns.

<p>(A-C) The tip bifurcation patterns with increasing spatial interval between bifurcation events. (D-F) The underlying spot patterns have a sparse distribution, with increasing number of spots, which correspond to the tip bifurcation patterns A-C. Parameters: <i>c</i> = 0.002, <i>μ</i> = 0.16, <i>ρ</i>_<i>A</i> = 0.03, <i>D</i>_<i>A</i> = 0.02, <i>v</i> = 0.04, <i>ρ</i>_<i>H</i> = 0.0001, <i>D</i>_<i>H</i> = 0.3, <i>c</i>₀ = 0.02, <i>γ</i> = 0.02, <i>D</i>_<i>S</i> = 0.06, <i>d</i> = 0.008, <i>e</i> = 0.1, <i>f</i> = 10, (A-C) <i>ε</i> = 1.5/1.0/0.7, (D-F) <i>S</i> = 0.320, <i>Y</i> = 0.185; <i>S</i> = 0.352, <i>Y</i> = 0.248; <i>S</i> = 0.395, <i>Y</i> = 0.313. In 2D patterns, black color indicates high concentration while gray indicates low.</p

Turing spot patterns underlying the branching patterns.

<p>(A) For the tip bifurcation pattern (Aa), the underlying Turing pattern is a spot distribution (Ab), which appears at the point of the cell differentiation trajectory (Ac, green curve) within the crescent-shaped Turing region (Ac, gray shadow region). (B) Same as (A) but for the side branching pattern. The Turing pattern underlying the side branching pattern is also a spot distribution. (Ca) The growth of tip bifurcation in the tip bifurcation pattern (Aa) in 3D form. (Cb) The tip bifurcation pattern (Aa) in 3D form. (Cc) The underlying spot pattern (Ab) in 3D form. (Da) The growth of side branching in the side branching pattern (Ba) in 3D form. (Db) The side branching pattern (Ba) in 3D form. (Dc) The underlying spot pattern (Bb) in 3D form. Parameters: <i>c</i> = 0.002, <i>μ</i> = 0.16, <i>ρ</i>_<i>A</i> = 0.03, <i>D</i>_<i>A</i> = 0.02, <i>v</i> = 0.04, <i>ρ</i>_<i>H</i> = 0.0001, <i>D</i>_<i>H</i> = 0.3, <i>c</i>₀ = 0.02, <i>γ</i> = 0.02, <i>D</i>_<i>s</i> = 0.06, <i>d</i> = 0.008, <i>e</i> = 0.1, <i>f</i> = 10, (Aa) <i>ε</i> = 1.0, (Ba) <i>ε</i> = 0.06, (Ab) <i>S</i> = 0.352, <i>Y</i> = 0.248, (Bb) <i>S</i> = 0.679, <i>Y</i> = 0.510. In the 2D patterns, black color indicates a high concentration, while gray indicates a low concentration. In the 3D patterns, red indicates a high concentration, while blue indicates a low concentration.</p

Comparison of the concentration gradients in different Turing patterns.

<p>(A) Spot pattern. (B) Stripe pattern. (C) Hole pattern. (D-F) The Turing patterns shown in 3D form corresponding to patterns A-C. The spot patterns exhibit the highest concentration gradient among the Turing patterns. The Turing patterns were generated by the activator-inhibitor model[<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0174946#pone.0174946.ref004" target="_blank">4</a>] with a saturation of activator production.</p

Migration of activator peaks in the transversal direction.

<p>(<b>a</b>) snapshot of YS domain (the YS domain is growing over time as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0102718#pone-0102718-g007" target="_blank">figure 7</a>). Morphogen concentration is denoted as z-axis height. Substrate has relatively low values inside the growing rectangle, and relatively high outside the growing rectangle. Y equals to 1.0 inside the rectangle and equal to 0.0 outside that rectangle. The initial rectangular is 5 space steps wide by 10 space steps long. The length of the rectangular increases one space step every 10,000 time steps. Space step dx = 0.3, time step dt = 0.4dx². (<b>b, c</b>) profile of S along the dotted line as shown in panel a. The high/low value of S profile is 1.0/0.6 and 1.0/0.4 in panel b and c. Activator peaks migrate out of the YS domain in a left-right order and a symmetrical manner under condition b and c respectively.</p