# 10.4: A phase space example - Biology

For an example of finding equilibria and stability, consider two competing species with intrinsic growth rates r1 = 1.2 and r2= 0.8. Let each species inhibit itself in such a way that s1,1 =−1 and s2,2 =−1, let Species 2 inhibit Species 1 more strongly than it inhibits itself, with s1,2 =−1.2, and let Species 1 inhibit Species 2 less strongly than it inhibits itself, with s2,1 =−0.5. The question is, what are the equilibria in this particular competitive system, and what will their stability be?

First, there is an equilibrium at the origin (0,0) in these systems, where both species are extinct. This is sometimes called the “trivial equilibrium,” and it may or may not be stable. From Table 10.2.1, the eigenvalues of the equilibrium at the origin are r1 and r2 —in this case 1.2 and 0.8. These are both positive, so from the rules for eigenvalues in Box 10.2.1, the equilibrium at the origin in this case is unstable. If no individuals of either species exist in an area, none will arise. But if any individuals of either species somehow arrive in the area, or if both species arrive, the population will increase. This equilibrium is thus unstable. It is shown in the phase space diagram of Figure (PageIndex{1}), along with the other equilibria in the system.

box (PageIndex{1}) calculated results for the sample competitive system

 Equilibrium Coordinates Eigenvalues Condition Origin (0,0) (1.2,0.8) Unstable Horizontal axis (1.2,0) (-1.2,0.2) Unstable Vertical axis (0,0.8) (-0.8,0.24) Unstable Interior (0.6,0.5) (-0.123,-0.977) Stable

On the horizontal axis, where Species 2 is not present, the equilibrium of Species 1 is N'1 = −r1/s1,1 = 1.2. That is as expected—it is just like the equilibrium of N' = −r /s for a single species—because it indeed is a single species when Species 2 is not present. As to the stability, one eigenvalue is −r1, which is −1.2, which is negative, so it will not cause instability. For the other eigenvalue at this equilibrium, you need to calculate q from Table 10.2.1. You should get q =−0.2, and if you divide that by s1,1, you should get 0.2. This is positive, so by the rules of eigenvalues in Box 10.2.1, the equilibrium on the horizontal axis is unstable. Thus, if Species 1 is at its equilibrium and an increment of Species 2 arrives, Species 2 will increase and the equilibrium will be abandoned.

Likewise, on the vertical axis, where Species 1 is not present, the equilibrium of Species 2 is N2 = −r2/s2,2 = 0.8. Calculate the eigenvalues at this equilibrium from Table 10.2.1 and you should get p=−0.24, and dividing by s2,2give eigenvalues of −0.8 and 0.24. With one negative and the other positive, by the rules of eigenvalues in Box 10.2.1 the equilibrium on the vertical axis is also unstable.

Finally, for the fourth equilibrium—the interior equilibirum where both species are present—calculate a, b, and c from the table. You should get a =−0.4, b =−0.44, and c=−0.048. Now the interior equilibrium is N'1 = p/a = 0.6 and N'2 = q/a = 0.5.

But is it stable? Notice the formula for the eigenvalues of the interior equilibrium in Table 10.2.1, in terms of a, b, and c. It is simply the quadratic formula! This is a clue that the eigenvalues are embedded in a quadratic equation, ax2 + bx + c = 0. And if you start a project to derive the formula for eigenvalues with pencil and paper, you will see that indeed they are. In any case, working it out more simply from the formula in the table, you should get −0.123 and −0.977. Both are negative, so by the rules of Box 10.2.1 the interior equilibrium for this set of parameters is stable.

As a final note, the presence of the square root in the formula suggests that eigenvalues can have imaginary parts, if the square root covers a negative number. The rules of eigenvalues in Box 10.2.1 still apply in this case, but only to the real part of the eigenvalues. Suppose, for example, that the eigenvalues are (frac{-1pmsqrt{-5}}{2},=,-0..5pm,1.118i). These would be stable because the real part, −0.5, is negative. But it turns out that because the imaginary part, (pm,1.118i), is not zero, the system would cycle around the equilibrium point, as predator–prey systems do.

In closing this part of the discussion, we should point out that eigenvectors and eigenvalues have broad applications. They reveal, for instance, electron orbitals inside atoms (right), alignment of multiple variables in statistics, vibrational modes of piano strings, and rates of the spread of disease, and are used for a bounty of other applications. Asking how eigenvalues can be used is a bit like asking how the number seven can be used. Here, however, we simply employ them to evaluate the stability of equilibria.

Program (PageIndex{1}) Sample program in R to generate a phase space of arrows, displaying the locations of the beginning and ends of the arrows, which are passed through a graphics program for display. The ‘while(1)’ statement means “while forever”, and is just an easy way to keep looping until conditions at the bottom of the loop detect the end and break out.

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## Cell Differentiation Examples

### In Animals

After the process of fertilization in animals, a single-celled organism called the zygote is formed. The zygote is totipotent, and will eventually become an entire organism. Even the largest animal on Earth, the blue whale, starts as a single cell. The complex tissues and organ systems, which are completely different in their form and function, all come from the zygote. The process of cell differentiation starts early within the organism. By the time the gastrula has formed, the cells have already started expressing various portions of the DNA.

As the systems continue to form, many of the stem cells lose their totipotency, themselves undergoing cell differentiation. This allows for faster production of specialized cells, which the growing organism needs to sustain its growth and enter the world with success. Through cell differentiation, tissues as different as brain tissue and muscle are formed from the same single cell.

### In Plants

While the plant lifecycle sometimes seems alien and complex, the process of cell differentiation is very similar. While there are different hormones involved, all plants also develop from a single cell. A seed is simply a protective housing for the zygote, which also provides a food supply. It is very similar to an egg in the animal world. The zygote inside undergoes cell division, and becomes a small embryo. Development is halted, as the seed is distributed into the world.

After winter, or anytime the environment is prime, the seed will soak up moisture and restart the process of development. The embryo will begin to form two meristems. A meristem is a unique portion of stem cells, which undergo cell differentiation as they grow outward. One will grow towards the surface, while the other will become the roots.

In the roots a layer of cells forms around the meristem, forming the root cap. This layer of cells sloughs off as the roots move through the soil, and are consistently replaced by the meristem. On the inside of the meristem, cell differentiation happens in a different direction. The hormones and environment here directs the cells to become vascular tissue and supporting cells. These will eventually carry water and nutrients to the top of the plant.

On the surface, the meristem acts in a similar way. As it divides upward, it creates both inward and outward cells. The inward cells undergo a differentiation similar to that of the roots, creating more vascular tissue. On the outside, the cells undergo cell differentiation into stems and leaves. These are equivalent to the different organs of animals, and are as different from the starting cells as animal cells. If you aren’t convinced, pick up an acorn and compare it to the massive tree it will become. Not only is it vastly smaller, it also contains completely different cell types. This can accounted for through the process of cell differentiation.

## Results

We search for symmetries in the E. coli transcriptional regulatory network [most updated compilation at RegulonDB (11)] where nodes are genes and a directed link represents a transcriptional regulation (SI Appendix, section III).

A directed link from a source gene i to a target gene j in a transcriptional regulatory network represents a direct interaction where gene i encodes for a transcription factor that binds to the binding site of gene j to regulate (activate or repress) its expression. Such a link represents a regulatory “message” sent by the source to the target gene using the transcription factor as a “messenger.” This process defines the “information flow” in the system which is not restricted to two interacting genes, but it is transferred to different regions within the network that are accessible through the connecting pathways. The information arriving to a gene contains the entire history transmitted through all pathways that reach this gene. We formalize this process of communication between genes with the notion of “input tree” of the gene. In a network G = ( N G , E G ) with N G nodes and E G directed edges, for every gene i ∈ N G there is a corresponding input tree, denoted as T i , which is the tree of all pathways of G ending at i. More precisely, T i is a rooted tree with a selected node i at the root, such that every other node j in the tree represents the initial node of a path in the network ending at i.

Next, we analyze the input trees in the E. coli subcircuit shown in Fig. 1A regulated by gene cpxR which regulates its own expression (via an autoregulation activator loop) and also regulates other genes as shown in Fig. 1A. Gene cpxR is not regulated by any other transcription factor in the network, so we say that this gene forms its own “strongly connected component” see below. Therefore, it is an ideal simple circuit to explain the concept of fibration.

Definition of input tree, symmetry fibration, fiber, and base. (A) The circuit controlled by the cpxR gene regulates a series of fibers as shown by the different-colored genes. The circuit regulates more genes represented by the dotted lines which are not displayed for simplicity. The full lists of genes and operons in this circuit are in SI Appendix, Table VI, ID nos. 27, 28, and 54. (B) The input tree of representative genes involved in the cpxR circuit showing the isomorphisms that define the fibers. For each fiber, we show the number of paths of length i − 1 at every layer of the input tree, a i , and its branching ratio n. (C) Isomorphism between the input trees of baeR and spy. The input trees are composed of an infinite number of layers due to the autoregulation loop at baeR and cpxR. How does one prove the equivalence of two input trees when they have an infinite number of levels? A theorem proved by Norris (26) demonstrates that it suffices to find an isomorphism up to N − 1 levels, where N is the number of nodes in the circuit. Thus, in this case, two levels are sufficient to prove the isomorphism. (D) Symmetry fibration ψ transforms the cpxR circuit G into its base B by collapsing the genes in the fibers into one. (E and F) Symmetry fibration of the fadR circuit (E) and its isomorphic input trees (F). Full list of genes in this circuit appears in SI Appendix, Table VI, ID nos. 3, 4, and 58. (G) Symmetric genes in the fiber synchronize their activity to produce the same activity levels. We use the mathematical model of gene regulatory kinetics from ref. 8 (sigmoidal interactions lead to qualitatively similar results) to show the synchronization inside the fiber baeR-spy when the fiber is activated by its regulator cpxR. Note that cpxR does not synchronize with the fiber.

### Input Tree Representation.

In practice, the input tree of a gene is constructed as follows (SI Appendix, section IV.A). Consider the circuit in Fig. 1A. The input tree of gene spy depicted in Fig. 1B starts with spy at the root (first layer). Since this gene is upregulated by baeR and cpxR, then the second layer of the input tree contains these two pathways of length 1 starting at both genes. Gene baeR is further upregulated by cpxR and by itself through the autoregulation loop and cpxR is also autoregulated. Thus, the input tree continues to the third layer taking into account these three possible pathways of length 2 from the source gene to the spy gene. The procedure now continues, and since there are loops in the circuit, the input tree has an infinite number of layers.

The input tree formalism is a powerful framework to search for symmetries in information-processing networks, in that it replaces the canonical notion of a single trajectory with the set of all possible “histories” from an initial to a final state of the network, and this makes, in practice, it reasonably straightforward to “guess” a type of symmetry which is not apparent in the classical network framework. Based on results from refs. 13 ⇓ ⇓ –16, we show in Symmetry Fibration Leads to Synchronization that if two input trees have the same “shape,” then the genes at the root of the input trees synchronize their activity (17 ⇓ ⇓ ⇓ ⇓ ⇓ –23), even though their input trees are made of different genes. This informal notion of equivalence is formalized by isomorphisms. An isomorphism between two input trees is a bijective map that preserves the topology of the input trees including the type of links. Specifically, a map τ : T → T ′ is an isomorphism if and only if for any pair of nodes a and b of T connected by a link, the pair of nodes τ ( a ) and τ ( b ) of T ′ is connected by the same type of link (SI Appendix, section IV.B). In practice, this means that isomorphic input trees are “the same” except for the labeling of the nodes. Genes with isomorphic input trees are symmetric and synchronous. We quantify this result, next, by introducing the concept of symmetry fibration (13).

### Symmetry Fibration of a Network.

The set of all input tree isomorphisms defines the symmetries of the network, which can be described by a “Grothendieck fibration” (12). The original Grothendieck definition of fibration is between categories (12), so the passage to a definition of fibration between graphs requires one to associate a category with a graph and rephrase Grothendieck’s definition in elementary terms. Different categories may be associated with a graph, giving rise to different notions of fibrations between graphs. The notion of fibration that we use henceforth has been introduced in computer science as a “surjective minimal graph fibration” (13, 15).

In general, a graph fibration G = ( N G , E G ) is any morphism ψ : G → B [1] that maps G to a graph B = ( N B , E B ) (with N B nodes and E B edges) called the “base” of the graph fibration ψ (SI Appendix, section IV.C). In this work we consider a surjective minimal graph fibration (13) which is a graph fibration ψ that maps all nodes with isomorphic input trees, comprising a “fiber,” to a single node in B, thus producing the minimal base of the network. In this case, the base B consists of a graph where all genes in a fiber have been collapsed into one representative node by the minimal fibration. Thus, a surjective minimal graph fibration, hereafter called symmetry fibration for the sake of lexical convenience, leads to a dimensional reduction of the network into its irreducible components. Crucially, a symmetry fibration is a dimensional reduction that preserves the dynamics in the network as we show next.

### Symmetry Fibration Leads to Synchronization.

Next, we explain the connection between fibration and synchrony in a generality that is needed to justify our results following refs. 15 and 16. To describe the dynamical state of each gene in the transcriptional regulatory network, we first attach a phase space to each node in G = ( N G , E G ) by considering a map P : N G → M that assigns each node i ∈ N G to the phase space of the node denoted by the manifold M. For example, in a transcriptional regulatory network we assign to each gene i ∈ N G the phase space of real numbers M = R . Then, the state of each gene is described by x i ( t ) ∈ R , representing the expression level of the gene i at time t, which is typically measured by mRNA concentration of gene product. We then obtain the total phase space of G as the product P G = ∏ i ∈ N G P ( i ) .

The fibers partition the graph G into unique and nonoverlapping sets Π = < Π 1 , … , Π r >, such that Π 1 ∪ ⋯ ∪ Π r = G and Π k ∩ Π l = ∅ if k ≠ l (24). We denote i ∼ Π j when the input trees of i and j are isomorphic and belong to the same fiber Π k . That is, ∃ k ∣ i , j ∈ Π k and there exists a symmetry fibration that sends both nodes to the same node in the base, ψ ( i ) = ψ ( j ) . DeVille and Lerman (15) showed that symmetry fibrations induce robust synchronization in the system (theorem 4.3.1 in ref. 15). In particular, it was shown that if ψ is a symmetry fibration, then—by proposition 2.1.12 in ref. 15—there exists a map P ψ : P B → P G that maps the total phase space of the base B, named P B , to the total phase space of the graph G. This map creates a polysynchronous subspace of synchronized solutions in fibers: Δ Π = < x ∈ P G ∣ x i ( t ) = x j ( t ) w h e n e v e r ψ ( i ) = ψ ( j ) >, where each set of synchronous components of this subspace corresponds to a fiber in Π (lemma 5.1.1 in ref. 15 see also ref. 16). In other words, Δ Π is a polysynchronous subspace of P G , such that components x i , x j ∈ x synchronize (i.e., x i ( t ) = x j ( t ) ) whenever the symmetry fibration ψ sends them to the same node in B.

According to these results, we interpret synchronous genes to process the same information received through isomorphic pathways in the network, and, accordingly, we interpret a symmetry fibration as a transformation that preserves the dynamics of information flow since it collapses synchronous nodes in fibers (redundant from the point of view of dynamics) into a common base with identical dynamics to those of the fiber.

Synchronous nodes in a fiber induced by symmetry fibration correspond to the “minimal balanced coloring” in ref. 14. A balanced coloring assigns two nodes the same color only if their inputs, self-consistently, receive the same content of colored nodes, whence the term “balanced.” Thus, the flow of information arriving to genes in a fiber is analogous to a process of assigning a color to each gene such that each gene “receives” the colors from adjacent genes via incoming links and “sends” its color to the adjacent genes via its outgoing links. The nodes in a fiber have the same color symbolizing the fact that they synchronize. The nodes with the same color in the balanced coloring partition (14) correspond to fibers induced by symmetry fibrations (15). We use the minimal balanced coloring algorithm proposed in ref. 25 for the computation of minimal bases (24) to find fibers (SI Appendix, section V).

### Strongly Connected Components of the E. coli Network.

The input trees in the E. coli cpxR circuit are displayed in Fig. 1B. The input trees of baeR and spy are isomorphic and define the baeR-spy fiber (Fig. 1C). We call this circuit a feedforward fiber (FFF). The input tree of cpxR is not isomorphic to either baeR or spy, and therefore cpxR is not symmetric with these genes, but it is isomorphic to bacA, slt, and yebE forming another fiber. Likewise, genes ung, tsr, and psd are all isomorphic, composing another fiber (Fig. 1B). Fig. 1D shows the symmetry fibration ψ : G → B that collapses the genes in the fibers to the base B. Fig. 1E shows another example (of many) of a single connected component, fadR, and its corresponding isomorphic input trees (Fig. 1F), fibers, and base.

The dynamical state of a gene is encoded in the topology of the input tree. In turn, this topology is encoded by a sequence, a i , defined as the number of genes in each ith layer of the input tree (Fig. 1B). The sequence a i represents the number of paths of length i − 1 that reach the gene at the root. This sequence is characterized by the branching ratio n of the input tree defined as a i + 1 / a i → i → ∞ n , which represents the multiplicative growth of the number of paths across the network reaching the gene at the root. For instance, the input trees of genes baeR-spy (Fig. 1B) encode a sequence a i = i with branching ratio n = 1 representing the single (n = 1) autoregulation loop inside the fiber.

Beyond several single-gene strongly connected components like those shown in Fig. 1, we find that the E. coli network has other strongly connected components (in a strongly connected component, each gene is reachable from every other gene SI Appendix, section VI), three in total, which regulate more involved topologies of fibers. We find 1) a two-gene strongly connected component composed of master regulators crp-fis involved in a myriad of functions like carbon utilization (Fig. 2 A, Top), 2) a five-gene strongly connected component involved in the stress response system (SI Appendix, Fig. S7), and 3) the largest strongly connected component at the core of the network which is composed of genes involved in the pH system that regulate hydrogen concentration (Fig. 2B). Each of these three components regulates a rich variety of fiber topologies which are collapsed into the base by the symmetry fibration ψ : G → B , as shown in Fig. 2B.

Strongly connected components of the genetic network and synchronization of gene coexpression in the fibers in E. coli. (A, Top) Two-gene connected component of crp-fis. This component controls a rich set of fibers as shown. We also show the symmetry fibration collapsing the graph to the base. We highlight the fiber uxuR-lgoR which sends information to its regulator exuR and forms a 2-Fibonacci fiber φ 2 = 1.6180 . . , ℓ = 2 , as well as the double-layer composite a d d − o x y S = 0,1 ⊕ 1,1 . (A, Bottom) Coexpression correlation matrix calculated from the Pearson coefficient between the expression levels of each pair of genes in A, Top. Synchronization of the genes in the respective fibers is corroborated as the block structure of the matrix. (B) The core of the E. coli network is the strongly connected component formed by genes involved in the pH system as shown. This component supports two Fibonacci fibers: 3-FF and 4-FF and fibers as shown. Open colored circles indicate genes that are in fibers and also belong to the pH component.

### Fiber Building Blocks.

We find that the transcriptional regulatory network of E. coli is organized in 91 different fibers. The complete list of fibers in E. coli is shown in SI Appendix, section VII and Table VI and the statistics are shown in SI Appendix, Table I. Plots of each fiber are shown in Dataset S1. We find a rich variety of topologies of the input trees. Despite this diversity, the input trees present common topological features that allow us to classify all fibers into concise classes of fundamental “fiber building blocks” (Fig. 3 A and B). We associate a building block to a fiber by considering the genes in the fiber plus the external incoming regulators of the fiber plus the minimal number of their regulators in turn that are needed to establish the isomorphism in the fiber. When the fiber is connected to any external regulator, either via a direct link or through a path in the strongly connected component forming a cycle, then the genes in this cycle are considered part of the building block of the fiber, since such a cycle is crucial to establish the dynamical synchronization state (when there is more than one cycle, the shortest cycle is considered).

Classification of building blocks in E. coli. (A) Basic fiber building blocks. These building blocks are characterized by a fiber that does not send back information to its regulator. They are characterized by two integer fiber numbers: n , ℓ . We show selected examples of circuits and input trees and bases. The full list of fibers appears in SI Appendix, Table VI and Dataset S1. The statistical count of every class is in SI Appendix, Table I. Bottom example shows a generic building block for a general n-ary tree n , ℓ with ℓ regulators. (B) Complex Fibonacci and multilayer building blocks. These building blocks are more complex and characterized by an autoregulated fiber that sends back information to its regulator. This creates a fractal input tree that encodes a Fibonacci sequence with golden branching ratio in the number of paths a i versus path length, i − 1 . When the information is sent to the connected component that includes the regulator, then a cycle of length d is formed and the topology is a generalized Fibonacci block with golden ratio φ d as indicated. We find three such building blocks: 2-FF, 3-FF, and 4-FF. Bottom panel shows a multilayer composite fiber with a feedforward structure.

We find that the most basic input tree topologies can be classified by integer “fiber numbers” n , ℓ reflecting two features: 1) infinite n-ary trees with branching ratio n representing the infinite pathways going through n loops inside the base of the fiber and 2) finite trees representing finite pathways starting at ℓ external regulators of the fiber. The most basic fibers in E. coli have three values of n = 0,1,2 (Fig. 3A): 1) fibers with n = 0 loops, called star fibers (SF) 2) fibers with n = 1 loop, called chain fibers (CF) and 3) fibers with n = 2 loops, called binary-tree fibers (BTF). This classification does not take into account the types of repressor or activator links in the building blocks, which lead to further subclasses of fibers that determine the type of synchronization (fixed point, limit cycles, etc.) and thus the functionality of the fibers.

Fig. 3A shows a sample of dissimilar circuits that can be concisely classified by n , ℓ (full list in Dataset S1). For instance, the n = 0 SF class includes dissimilar circuits like a r c Z − y d e A = 0,1 , d c u C − a c k A = 0,2 , which is a bifan network motif (2), and generalizations with ℓ = 3 regulators like d c u R − a s p A = 0,3 (Fig. 3 A, Top). The main feature of these building blocks is that they do not contain loops and therefore the input trees are finite. The CF class contains n = 1 loop in the fiber and therefore an infinite chain in the input tree, like the autoregulated loop in the fiber t t d R = 1,0 . We note that while the input tree is infinite, the topological class is characterized by a single number n = 1 concisely represented in the base. Furthermore, a theorem proved by Norris (26) demonstrates that it suffices to test N G − 1 layers of the input trees to prove isomorphism, even though the input tree may contain an infinite number of layers. Adding one external regulator ( ℓ = 1 ) to this circuit converts it to the purine fiber p u r R = 1,1 which is an example of a FFF, like the baeR circuit in Fig. 1A. This circuit resembles a feedforward loop motif (2), but it differs in the crucial addition of the autoregulator loop at purR that allows genes purR and pyrC to synchronize. When another external regulator is added, we find the idonate fiber i d n R = 1,2 . More elaborated circuits contain two autoregulated loops and feedback loops featuring trees with branching ratio n = 2 .

### Fibonacci Fibers.

So far we have analyzed building blocks that receive information from the external regulators in their respective strongly connected components, but do not send back information to the external regulators. These topologies are characterized by integer branching ratios, n = 0,1,2 , as shown in Fig. 3A. We find, however, more interesting building blocks that also send information back to their regulators. These circuits contain additional cycles in the building blocks that transform the input trees into fractal trees characterized by noninteger fractal branching ratios. Notably, the building block of the fiber uxuR-lgoR that is regulated by the connected component crp-fis (Fig. 2) forms an intricate input tree (Fig. 3 B, Top) where the number of paths of length i − 1 is encoded in a Fibonacci sequence a i = 1, 3, 4, 7, 11, 18, 29, … characterized by the Fibonacci recurring relation a 1 = 1 , a 2 = 3 , and a i = a i − 1 + a i − 2 for i > 2 . This sequence leads to the noninteger branching ratio known as the golden ratio: a i + 1 / a i → i → ∞ φ = ( 1 + 5 ) / 2 = 1.6180 . . .

This topology arises in the genetic network due to the combination of two cycles of information flow. First, the autoregulation loop inside the fiber at uxuR creates a cycle of length d = 1 which contributes to the input tree with an infinite chain with branching ratio n = 1 . This sequence is reflected in the Fibonacci series by the term a i = a i − 1 . The important addition to the building block is a second cycle of length d = 2 between uxuR in the fiber and its regulator exuR: uxuRexuRuxuR. This cycle sends information from the fiber to the regulator and back to the fiber by traversing a path of length d = 2 that creates a “delay” of d = 2 steps in the information that arrives back to the fiber (Fig. 3 B, Top). This short-term “memory” effect is captured by the second term a i = a i − 2 in the Fibonacci sequence leading to a i = a i − 1 + a i − 2 and the golden ratio. We call this topology a Fibonacci fiber (FF).

This argument implies that an autoregulated fiber that further regulates itself by connecting to its connected component via a cycle of length d encodes a generalized Fibonacci sequence of order d defined as a i = a i − 1 + a i − d with generalized golden ratio φ d (Fig. 3 B, Top, fourth row). We find such a Fibonacci sequence in the evgA-nhaR fiber building block linked to the pH strongly connected components shown in Fig. 2B. This fiber contains an autoregulation cycle inside the fiber and also an external cycle of length d = 4 through the pH strongly connected component: evgAgad EgadXhnsevgA (Fig. 3 B, Top, third row). This topology forms a fractal input tree with sequence a i = a i − 1 + a i − 4 (sequence A003269 in ref. 27) and branching golden ratio φ 4 = 1.38028 . . . We call this topology 4-Fibonacci fiber, 4-FF. Generalized Fibonaccis appear inside strongly connected components, like the rcsB-adiY 3-FF in the pH system (Fig. 3 B, Top, second row). Likewise, if the network contains many cycles of varying length up to a maximum d, the Fibonacci sequence generalizes to a i = a i − 1 + a i − 2 + ⋯ + a i − 1 − d + a i − d , and the branching ratio satisfies d = − log ( 2 − φ d ) log φ d (28).

### Multilayer Composite Fibers.

Building blocks can also be combined to make composite fibers, like prime numbers or quarks can be combined to form natural numbers or composite particles like protons and neutrons, respectively. The ability to assemble fiber building blocks to make larger composites is important in that it helps to understand systematically higher-order functions of biological systems composed of many genetic elements. We discover a particular type of composite made up of two elementary building blocks that we name multilayer composite fiber. For instance, the double-layer add-oxyS fiber in the crp-fis connected component (Figs. 2A and 3 B, Bottom and ID 7 in SI Appendix, Table VI and Dataset S1) is a composite a d d − o x y S = 0,1 ⊕ 1,1 made of a series of genes composing a single fiber of type 0,1 = a d d , d s b G , g o r , g r x A , h e m H , o x y S , t r x C that are regulated by two different transcription factors rbsR and oxyR that form another fiber of type 1,1 = r b s R , o x y R . This composite is of importance since it allows for information to be shared between two genes, for instance add and oxyS, which are not directly connected (in this case, separated by a distance in the network of length 2 from crp).

Composite fibers satisfy a simple engineering “sum rule”: elementary fibers are composed in series of fibers in a predefined order where the first layer is represented by an entry fiber (carrying transcription factors), and the last layer is formed by a terminator fiber of output genes (encoding enzymes), as shown in Fig. 3 B, Bottom. This multilayer composite fiber is biologically significant because genes in the output layer synchronize a genetic module that implements the same function even though the genes in the module are not directly connected and, indeed, can be at far distances in the network. Such functionally related modules could not be identified by modularity algorithms (29) which cluster nodes in modules of highly connected nodes.

We find that composite fibers are dominant in eukaryotes (yeast, mice, humans see Fibration Landscape across Biological Networks, Species, and System Domains). They resemble the building blocks of multilayered deep neural networks where each subsequent gene in the layer synchronizes despite the fact that nodes can be distant in the network. More generally, composite fibers with multiple layers streamline the construction of larger aggregates of fibration building blocks, performing more complex function in a coordinated fashion. These composite topologies complete the classification of input trees.

### Fibration Landscape across Biological Networks, Species, and System Domains.

To study the applicability of fibration symmetries across domains of complex networks we have analyzed 373 publicly available datasets (SI Appendix, section VIII). Full details of each network and results can be accessed on GitHub at https://github.com/makselab/fibrationData/blob/master/datasets.xlsx. The codes to reproduce this analysis are on GitHub at https://github.com/makselab (SI Appendix, section V). The full datasets are on GitHub at https://github.com/makselab/fibrationData/blob/master/rawData.zip. We analyze biological networks spanning from transcriptional regulatory networks, metabolic networks, cellular processes networks and signaling pathways, disease networks, and neural networks. We span different species ranging from Arabidopsis thaliana, E. coli, Bacillus subtilis, Salmonella enterica, Mycobacterium tuberculosis, Drosophila melanogaster, Saccharomyces cerevisiae (yeast), and Mus musculus (mouse) to Homo sapiens (human). The topological fiber numbers n , ℓ allow us to systematically classify fibers across the different domains in a unifying way. We find that fibration symmetries are found across all biological processes and domains. The fiber distributions for each type of biological network calculated by summing over the studied species are displayed in Fig. 4A and the fiber distributions for each species calculated over the type of biological networks are shown in Fig. 4B. Our analysis allows us to investigate the specific attributes and commonalities of the fiber building blocks inside and across biological domains. We find a varied set of fibers that characterize the biological landscape. Certain features of the fiber number distribution are visible in the transcriptional networks in Fig. 4A. For instance, a tail with ℓ is seen in the n = 0 class as well as in the n = 1 class. Across species (Fig. 4B), bacteria like E. coli or B. subtilus display a majority of n = 0 building blocks, while higher-level organisms like yeast, mice, and humans display a majority of more complex building blocks like multilayers and Fibonaccis.

Fibration landscape across domains and species. (A) Fibration landscape for biological networks. Shown is the total number of fiber building blocks across five types of biological networks analyzed in the present work. The count includes the total number of fibers in the networks of each biological type considering all species analyzed for each type (SI Appendix, Table IV). (B) Fibration landscape across species. Shown is the count of fibers across each analyzed species. Each panel shows the count over the different types of biological networks (E. coli contains only the transcriptional network see SI Appendix, Table IV). (C) Fibration landscape across domains. Shown is the count of fibers across the major domains studied. The biological domain panel is calculated over all networks and species in A and B. (D) Global fibration landscape. Shown is the cumulative count of fibers in all domains in C. The cumulative count represents the total number of fibers per network of 1 0 4 nodes. Specifically, the quantity is calculated as the total number of fibers divided by the total number of nodes in all networks per domain multiplied by 1 0 4 .

To test the existence of symmetry fibrations across other domains we extend our studies to complex networks beyond biology ranging from social, infrastructure, internet, software, and economic networks to ecosystems (details of datasets in SI Appendix, section VIII). Fig. 4C shows the obtained fiber distributions for each domain. A normalized comparison across domains is visualized in Fig. 4D, showing the cumulative number of fibers over all domains and species per network size of 10 4 nodes. The results support the applicability of the concept of symmetry fibration beyond biology to describe the building blocks of networks across all domains.

### Gene Coexpression and Synchronization via Symmetry Fibration.

We have shown in Symmetry Fibration Leads to Synchronization that fibers in networks determine cluster synchronization in the dynamical system. In a gene regulatory network, symmetric genes in a fiber synchronize their activity to produce gene coexpression levels that sustain cellular functions. We corroborate this result numerically in Fig. 1G in the particular example of the baeR-spy FFF in E. coli, and this result applies to all fibers, irrespective of the dynamical system law.

To exemplify the synchronization in fibers, we consider the dynamics in the composite fiber a d d − o x y S = 0,1 ⊕ 1,1 depicted in Figs. 2A and 3 B, Bottom, which is composed of autoregulator 1 = c r p , and two layers of fibers: 2 = r b s R , 3 = o x y R and 4 = a d d , 5 = o x y S (we consider here a reduced fiber for simplicity, and we add the autoregulator to crp to the building block for completeness). Graph G = < N G , E G >consists of N G = < 1,2,3,4,5 >, E G = < 1 → 1,1 → 2,1 → 3,2 ⊣ 2,3 ⊣ 3,2 → 4,3 → 5 >(⊣ refers to repressor and → to activation), and a five-dimensional total phase space P G = R 5 with state vector X ( t ) = < x 1 ( t ) , x 2 ( t ) , x 3 ( t ) , x 4 ( t ) , x 5 ( t ) >describing the expression levels of each gene’s product (e.g., mRNA concentration).

The symmetry fibration ψ : G → B collapses the graph G into the base B = < N B , G B >, where N B = < a , b , c >and E B = < a → a , a → b , b ⊣ b , b → c >. The symmetry fibration acts on the nodes ψ ( 1 ) = a , ψ ( 2 ) = ψ ( 3 ) = b , and ψ ( 4 ) = ψ ( 5 ) = c and on the edges ψ ( 1 → 1 ) = a → a , ψ ( 1 → 2 ) = ψ ( 1 → 3 ) = a → b , ψ ( 2 ⊣ 2 ) = ψ ( 3 ⊣ 3 ) = b ⊣ b , and ψ ( 2 → 4 ) = ψ ( 3 → 5 ) = b → c . Thus, the fibers partition the graph G as Π = < Π a , Π b , Π c >, where Π a = < 1 >, Π b = < 2,3 >, and Π c = < 4,5 >.

We represent the dynamics by two functions k ( x ) and g ( x ) modeling degradation and synthesis of gene product, respectively (9, 10). For example, k ( x ) can be modeled as a linear degradation term and g i ( x ) as a Hill function ( i = A , R , activation or repression) (9). We consider that multiple inputs are combined by multiplying functions g ( x ) , but any other way of combining inputs can be used. Then, the dynamics of the expression levels of the genes in the circuit are described by ref. 14: d x 1 d t = − k ( x 1 ) + g A ( x 1 ) d x 2 d t = − k ( x 2 ) + g A ( x 1 ) * g R ( x 2 ) d x 3 d t = − k ( x 3 ) + g A ( x 1 ) * g R ( x 3 ) d x 4 d t = − k ( x 4 ) + g A ( x 2 ) d x 5 d t = − k ( x 5 ) + g A ( x 3 ) . [2] The dynamics of the base are described by the state vector of the base: ( y a ( t ) , y b ( t ) , y c ( t ) ) with dynamical equations (16): d y a d t = − k ( y a ) + g A ( y a ) d y b d t = − k ( y b ) + g A ( y a ) * g R ( y b ) d y c d t = − k ( y c ) + g A ( y b ) . [3] If ( y a ( t ) , y b ( t ) , y c ( t ) ) is a solution for the base Eq. 3, then the map P ψ sends the phase space of this base to the phase space of the solutions in the graph G (16): x 1 ( t ) , x 2 ( t ) , x 3 ( t ) , x 4 ( t ) , x 5 ( t ) = P ψ y a ( t ) , y b ( t ) , y c ( t ) = y a ( t ) , y b ( t ) , y b ( t ) , y c ( t ) , y c ( t ) . [4] Therefore, the graph G sustains a polysynchronous subspace (see for instance motivating example 1.4 in ref. 15): Δ Π = ( x 1 , x 2 , x 3 , x 4 , x 5 ) ∈ R 5 ∣ x 1 ( t ) , x 2 ( t ) = x 3 ( t ) , x 4 ( t ) = x 5 ( t ) . [5] This result can be corroborated by simply plugging x 1 ( t ) , x 2 ( t ) , x 3 ( t ) = x 2 ( t ) , x 4 ( t ) , x 5 ( t ) = x 4 ( t ) into Eq. 2 to obtain a solution of the dynamics, implying the synchrony x 2 ( t ) = x 3 ( t ) in fiber Π b and x 4 ( t ) = x 5 ( t ) in fiber Π c . We note that the concept of sheaves and stacks might be useful to generalize the symmetry fibration framework to multiplex networks.

We test this gene synchronization with publicly available transcription profile experiments available from the literature. We use gene expression data profiles in E. coli compiled at Ecomics: http://prokaryomics.com (30). This portal collects microarray and RNA-seq experiments from different sources such as the NCBI Gene Expression Omnibus (GEO) public database (31) and ArrayExpress (32) under different experimental growth conditions. The data are also compiled at the Colombos web portal (33). The database contains transcriptome experiments measuring the expression level of 4,096 genes in E. coli strains over 3,579 experimental conditions which are described as strain, medium, stress, and perturbation. Raw data are preprocessed to obtain expression levels by using noise reduction and bias correction to normalize data across different platforms (30).

E. coli can adapt its growth to the different conditions that it finds in the medium. This adaptation is made by sensing extra and intracellular molecules and using them as effectors to activate or repress transcription factors. This implies that the different fibers are activated by specific experimental conditions. The Ecomics portal allows one to obtain those experimental conditions where a set of genes has been significantly expressed under a particular set of conditions. We perform standard gene expression analysis (http://colombos.net and ref. 33) of the expression levels in E. coli obtained under these conditions.

For a given set of genes in a fiber, we find the experimental conditions for which the genes have been significantly expressed by comparing the expression samples over the 1,576 different WT growth conditions. Following ref. 33, the experimental conditions are ranked with the inverse coefficient of variation (ICV) defined as ICV k = | μ k | / σ k , where μ k is the average expression level of the genes in the condition k and σ k is the SD. Following ref. 33, we select those conditions with I C V k > ⟨ ICV k ⟩ , i.e., where the average expression levels in the particular condition k are significantly higher than the SD. This score reflects the fact that, in a relevant condition, the genes show an increment of their expression above the individual variations caused by random noise. Details on the expression analysis can be found in ref. 33 and https://doi.org/10.1371/journal.pone.0020938.s001. Thus, we obtain expression levels organized by the relevant experimental conditions which are labeled according to the GEO database (31). From these data, we calculate the coexpression matrix using the Pearson correlation coefficient between the expression levels of two genes i and j in the relevant conditions for genes in a fiber. For off-diagonal correlations between genes in different fibers, we use the combined sets of conditions of both genes.

Results for the correlation matrix are shown in Fig. 2 A, Bottom for fibers regulated by the crp-fis strongly connected component. Gene expression is obtained for every gene, so we plot the correlation matrix calculated over each pair of genes. Genes that belong to the same operon are transcribed as a single unit by the same mRNA molecule, so these genes are expected to trivially synchronize (variations exist due to attenuators inside the operon). Thus, we group together these genes as operons in Fig. 2A to indicate this trivial synchronization. To test the existence of fiber synchronization we compare gene coexpression belonging to different operons. Fig. 2 A, Bottom shows that expression levels of the genes that belong to a fiber are highly correlated as predicted by the symmetry fibration. Genes that belong to different fibers show no significant correlations among them. In particular, there is no significant correlation between the expression of genes in a given fiber and the two master regulators crp and fis. This result is consistent with the fibration symmetry and occurs despite the fact that both crp and fis directly regulate all genes in the studied fibers. We find some off-diagonal weak correlations between fibers (e.g., malI), probably indicating missing links or missing regulatory processes that produce extra synchronizations. Some genes present weak correlations inside fibers (e.g., cirA), indicating weak symmetry breaking probably from asymmetries in the strength of binding rate of transcription factors or input functions, effects that are not considered in the topological view of the input trees and can lead to desynchronization inside the fiber.

## 10.4: A phase space example - Biology

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Carbon dioxide can exist as a solid, liquid, or gas under specific temperatures and pressures. This dependence is mapped into a phase diagram, which includes three general features: regions, lines, and points.

Regions represent the temperature and pressure conditions for a single phase. At standard pressure, carbon dioxide can be solid or gaseous, depending on the temperature.

At pressures below 5.1 atm, if the temperature of solid carbon dioxide is raised, it will directly transition to the gaseous phase without passing through the liquid form.

A transition through all three phases — solid to liquid and liquid to gas —  will occur at pressures above 5.1 atm.

The lines or curves that separate the regions denote the temperatures and pressures at which the phases on either side of the curve are in equilibrium.

For example, the point at 1 atm and 󔽖.5 °C, lies on the curve separating the solid and vapor phases, so a solid–vapor equilibrium exists under those conditions. Accordingly, this is called the sublimation curve.

Similarly, the liquid-gas equilibrium exists on the vaporization curve, and the solid-liquid equilibrium exists on the fusion curve. These curves are known more generally as phase boundaries.

At 5.1 atm and 󔽀.6 °C all three phases will coexist. This is the triple point of carbon dioxide.

At 73 atm and 31 °C, both liquid and vapor phases of carbon dioxide will coalesce into a single-phase supercritical fluid. This is the critical point of carbon dioxide.

In the region beyond the critical point no pressure or temperature change can convert a supercritical fluid to a gas or liquid.

The phase diagram of water has a few notable differences from that of carbon dioxide.

The fusion curve of carbon dioxide has a positive slope, while for water, the slope is negative. This is an atypical feature of water. Increasing the pressure favors a liquid-to- solid transition in carbon dioxide but a solid-to-liquid transition in water.

The higher pressure favors the denser solid form of carbon dioxide. In the case of water, the denser liquid form is favored

### 11.13: Phase Diagrams

A phase diagram combines plots of pressure versus temperature for the liquid-gas, solid-liquid, and solid-gas phase-transition equilibria of a substance. These diagrams indicate the physical states that exist under specific conditions of pressure and temperature and also provide the pressure dependence of the phase-transition temperatures (melting points, sublimation points, boiling points). Regions or areas labeled solid, liquid, and gas represent single phases, while lines or curves represent two phases coexisting in equilibrium (or phase change points). The triple point indicates conditions of pressure and temperature at which all three phases coexist. In contrast, a critical point indicates the temperature and pressure above which a single phase—whose physical properties are intermediate between the gaseous and liquid states—exists.

Figure 1. A typical phase diagram.

A phase diagram identifies the physical state of a substance under specified conditions of pressure and temperature. To illustrate the utility of these plots, consider the phase diagram of water, shown below.

Figure 2. Phase diagram of water.

A pressure of 50 kPa and a temperature of 󔼒 °C corresponds to the region of the diagram labeled “ice.” Under these conditions, water exists only as a solid. A pressure of 50 kPa and a temperature of 50 °C corresponds to the region where water exists only as a liquid. At 25 kPa and 200 °C, water exists only in the gaseous state. Curve BC is the liquid-vapor curve separating the liquid and gaseous regions of the phase diagram and provides the boiling point for water at any pressure. For example, at 1 atm, the boiling point is 100 °C. Notice that the liquid-vapor curve terminates at a temperature of 374 °C and a pressure of 218 atm, indicating that water cannot exist as a liquid above this temperature, regardless of the pressure. The physical properties of water under these conditions are intermediate between those of its liquid and gaseous phases. This unique state of matter is called a supercritical fluid. The solid-vapor curve labeled AB, indicates the temperatures and pressures at which ice and water vapor are in equilibrium. These temperature-pressure data pairs correspond to the sublimation, or deposition, points for water.

The solid-liquid curve labeled BD shows the temperatures and pressures at which ice and liquid water are in equilibrium, representing the melting/freezing points for water. Note that this curve exhibits a slight negative slope, indicating that the melting point for water decreases slightly as pressure increases. Water is an unusual substance in this regard, as most substances exhibit an increase in melting point with increasing pressure. The point of intersection of all three curves—labeled B—is the triple point of water, where all three phases coexist in equilibrium. At pressures lower than the triple point, water cannot exist as a liquid, regardless of the temperature.

Consider the phase diagram for carbon dioxide as another example.

Figure 3. Phase diagram of carbon dioxide.

The solid-liquid curve exhibits a positive slope, indicating that the melting point for CO2 increases with pressure as it does for most substances. Notice that the triple point is well above 1 atm, indicating that carbon dioxide cannot exist as a liquid under ambient pressure conditions. Instead, cooling gaseous carbon dioxide at 1 atm results in its deposition into the solid-state. Likewise, solid carbon dioxide does not melt at 1 atm pressure but instead sublimes to yield gaseous CO2. Finally, the critical point for carbon dioxide is observed at a relatively modest temperature and pressure in comparison to water.

## Phase Space vs. Position Space

It has been argued that these definitions apply only to "Phase Space," and that Phase Space is entirely different than the normal Position Space that we know which uses the x, y, and z coordinates. It has been argued that Phase Space does not contain the Position Space coordinates.

In truth, Phase Space is merely Position Space with additional dimensions. We read a definition from Physics for Degree Students B.Sc Second Year (Archive):

10.4 Phase Space   “  A combination of the position space and momentum space is known as phase space. Thus phase space has six dimensions. A point in phase space is, therefore, completely specified by six coordinates x, y, z, px, py, pz. Complete information about any particle in a dynamic system can be obtained from a knowledge of these six co-ordinates which completely determine its position as well as momentum. As there are n particles a knowledge of 6n co-ordinates gives complete information regarding position and momentum of all then particles in the phase space for a dynamic system. The concept of phase space is very useful while dealing with dynamical systems actually existing in nature.  ”

Scrolling up to section 10.2 we read the definitions of Position Space, verifying that it is the normal x, y, z position coordinates:

“  The three dimensional space in which the location of a particle is completely given by the three position coordinates, is known as position space.  ”

Thus, if the geometry of Phase Space is preserved with a Symplectic Integrator, the geometry of Position Space is also preserved.

### Figure Eight Example

Compare this figure eight three body problem in Phase Space and Position Space:

Compare the above to the Figure Eight Three Body Problem solution in Position Space, plotted on the x and y.

Plotted on the x and y, and looks surprisingly like the Phase Space version.

It looks the same because it is the same. Phase Space is merely Position Space with detail in additional dimensions which represents momentum. The shape of an orbit represents geometry, as well as energy loss/gain.

## 99391987

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### 99391987

99391987 Grundfos Pump, CRNE 10-4 A-FGJ-A-E-HQQE Vertical Multistage Centrifugal, 2" x 2" ANSI Flange Connection, Basic Version, 4 Stages, 3 HP, 200-240 Volt, 60 Hz, 3 Ph (Three Phase), 3461 RPM (Rated), TEFC - Totally Enclosed Fan Cooled Motor Enclosure, EPDM Shaft Seal. 250° F Max Op Temp. Materials: Basic Version.

 Bearing Material SIC Capacities (GPM) 53.3 GPM Connections / Discharge (Inches) 2" x 2" Connection Type ANSI Flange Discharge (Inches) 2" Heads To (Feet) 150.3 Feet HP 3 Hz 60 Impeller Material Stainless Steel Maximum Pressure (PSI) 363 PSI Model Number CRNE10-04 Motor Enclosure Type TEFC - Totally Enclosed Fan Cooled
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## 4 Major Phases of the Cell Cycle (With Diagram)

The G1 phase is set in immediately after the cell division. It is characterised by a change in the chromosome from the condensed mitotic state to the more extended interphase state and by a series of metabolic events leading to initiation of DNA replication. G, phase of cell cycle varies in length from cell to cell within the same cell population.

The length of this phase is more than the other three phases. This period represents in general 25 to 40% of the generation time of a cell. The cause of variability in G1 is not known, although there is an evidence to suggest, that the protein content of the cell possibly determines the length of G1.

The G1 is a very significant phase of cell cycle in the sense that the cells which stop dividing normally remain arrested in this phase. Within G1 phase, there is a definite check point at which DNA synthesis is initiated and once the biochemical events associated with that point have occurred the cell proceeds towards division.

There are specific molecular signals at that point which decide whether a cell has to continue its proliferation or it has to specialize into a permanent or non-dividing cell.

During G1, phase the chromatin fibres become slender, less coiled and fully extended and more active for transcription. The nucleus appears under light microscope to contain a network of delicate fibres. The transcriptional activities result in the synthesis of rRNA, tRNA, mRNA and a series of molecules required for the initiation of DNA synthesis.

The events which lead to the initiation of DNA synthesis include synthesis of enzymes and other proteins required for DNA synthesis.

#### Cell Cycle: Phase # 2. S Phase:

After G1, phase there comes the S phase. Biochemically, it is a phase of active DNA and histone synthesis. During S period doubling of the slender fully extended chromosomes takes place which is accomplished by doubling of DNA and the associated proteins in the chromosomes.

The replication of DNA is semiconservative and discontinuous. J.H. Taylor in 1957 labelled the root tip cells of broad bean with thymidine labelled with tritium ( 3 H ).

The labelled cells were found to contain labelled chromatin during S period in auto-radiographs. Such labelled cells were allowed to go through one more division cycle in absence of label and then it was found that at any point along each chromosome, only one chromatid was labelled.

This indicated semiconservative replication of DNA. In bacteria there is a single replication point and replication of DNA proceeds bidirectionally from that point.

In eukaryotes, each chromosome has a number of replication units (RUs) or replicons, each of which has a specific origin and two termini for replication process. Electron microscopy and autoradiographic studies have indicated that the replication of DNA begins by the opening of a “bubble” in DNA resulting into two forks.

These forks migrate bidirectionally as the DNA is synthesised until they reach termination points. There are specific nucleotide sequences which mark the initiation and termination points. Careful examination of electron micrographs of replicating chromosomes shows the spacing between the replication units which, in a wide range of eukaryotes, ranges from 7 to 100 mm (30,000 to 300,000 base pairs).

After the termination process, the adjacent replication forks 166 Cytology are fused. Since DN A replication is dependent on protein and RNA synthesis for the overall replication of chromosome, it is necessary that new proteins must be synthesised.

Although some of the histone protein is synthesised during G1, phase, most of it is synthesised during S phase. Further, synthesis of ribosomal RNA must continue from G1, to S phase if DNA synthesis is to start.

The eukaryotic chromosomes consist of DNA-histones complexes, called nucleosomes. When the DNA is replicated and new histones are synthesised the two are complexed to form nucleosomes.

The distribution of old and newly synthesised histones in the nucleosomes of newly replicated DNA is not clearly understood but it is clear that the old octamers of histones are conserved from generation to generation and the octamers of newly synthesised histones are used at the replication forks to form new nucleosomes.

#### Cell Cycle: Phase # 3. G2 Phase:

G2 phase follows the S phase. The metabolic significance of this phase is not fully understood as yet. This phase represents to some 10 to 25% of the generation time of the cell. In G2, the chromosome consists of two strands or chromatids. The cells that are arrested at the transition between S and G2 will have twice the usual amount of DNA. Tetraploid nuclei are occasionally found in various tissues, as for example, in liver cells, heart muscles etc.

The critical event at G2, check point is concerned with the condensation of chromatin. The condensation process involves high order folding of the chromatin fibre but the mechanism for this is poorly understood. The chromatin condensation is associated with accumulation of a cytoplasmic substance.

The existence of chromatin condensing factor has been demonstrated by R.T. Johnson and P.N. Rao (1970, 74) by using Sendai virus to fuse He La cells (a type of cell derived from human cancer cell) at different stages of cell cycle. When a mitotic cell was fused with a cell at G2 phase, chromatin condensation took place at once in G2 nucleus, producing normal looking chromosomes.

When a mitotic cell was fused with a cell at G1, phase chromatin condensation was demonstrated in G1, nucleus also. Similarly, condensation of chromatin in S nucleus was also observed when mitotic cell was fused with a cell at S phase, although multiple fragments of condensed chromatin instead of complete chromosomes were seen.

The chromatin condensation is thought to be associated with, and perhaps due to, phosphorylation of histone H.

The syntheses of RNAs and proteins occur during most of the cell cycle, but the two processes may be inhibited during S phase and almost completely suppressed during late prophase. Another important event that takes place during G2 period is the synthesis of the proteins tubulin which is later polymerized to produce microtubules that make up spindle apparatus during M phase.

#### Cell Cycle: Phase # 4. M Phase:

M phase follows G2 phase. During this phase the cell divides into two daughter cells. The chromosomes are duplicated during interphase and they are distributed to the progeny cells by division process. After M phase, the resulting daughter cells then enter the G1, phase of next cell cycle. The cells which after completion of mitosis do not enter G1, are referred to as G0 cells.

During M phase, RNA synthesis stops at late prophase and resumes at telophase. The protein synthesis drops drastically and during that period RNA synthesis also stops. These changes in the synthetic activity may be attributed to the non-availability of transcription sites owing to highly condensed state of the chromosomes. If no mRNA is available to ribosomes, protein synthesis cannot take place.

The nucleolus and nuclear wall break down goes side by side with the inhibition of RNA and protein synthesis. These structures are reformed in daughter nuclei in late telophase when the synthesis of RNA and proteins resume. The molecular signals for these changes are not known. The behaviour of chromosomes during different phases of cell cycle, as illustrated by De Robertis et al. (1975) is shown in (Fig. 11.1).

## Consider This Phase Diagram For Carbon Which Phases Are Present At The Lower Triple Point

Which phases are present at the lower triple point. Which phases are present at the upper triple point.

10 4 Phase Diagrams Chemistry

### Graphite 10 gas diamond 1 liquid starting from the lower triple point what actions would produce liquid carbon.

Consider this phase diagram for carbon which phases are present at the lower triple point. We are finally in a position to define the triple point it is just the point at which the two phase equilibrium lines intersect. Consider this phase diagram for carbon which phases are present at the lower triple point what do the triple point and critical point on a phase the triple point of a phase diagram is the location where the solid liquid and gas phases meet it is the temperature and pressure at which a given substance can assume any of the 3 usual phases of. Label the diamond phase.

A special critical point where solid liquid and vapor coexiste is a triple point. It is worth considering the implications of that difference. Diamond graphite gas liquid which phase is stable at 105 atm and 1000 k.

On the phase diagram label the graphite phase. Which phases are present at the lower triple pointa. Problems and solutions book.

Consider this phase diagram for carbon. B graphite is the most stable phase of carbon at normal conditions. Raise the temperature raise the pressure lower the pressure lower the temperature.

For most substances the slope of the liquid solid phase equilibrium curve is positive. All substances except helium have triple points. Points on a phase diagram that meet are called critical points.

The other critical points are where there are three phasescrystal structures but not solid liquid and vapor as all three. Graphite gas diamond liquid which phase is stable at 105 atm and 1000 k. For water the slope is negative.

Consider this phase diagram for carbon. For a given sample of carbon dioxide co2 you increase the temperature from 80c to 0c and decrease the pressure from 30 atm to 5 atm. Gas liquid diamond graphite starting from the lower triple point what actions would produce liquid carbon.

Refer to the phase diagram for carbon dioxide in problem set 60. Chemistry end of chapter exercises. C if graphite at normal conditions is heated to 2500 k while the pressure is increased to 10 10 pa it is converted into diamond.

Consider this phase diagram for carbon. Liquid which phases are present at the lower triple pointa. Phase diagrams in the chemistry.

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Consider This Phase Diagram For Carbon Which Phases Are Present At The Lower Triple Point A Diamondb Graphitec Gasd Liquid

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## What Relationship Exists Between Solubility And Temperature For Most Of The Substances Shown - 10 4 Phase Diagrams Chemistry / However, this is not the case for sodium sulfate above 30ºc where the solubility then begins to decrease.

What Relationship Exists Between Solubility And Temperature For Most Of The Substances Shown - 10 4 Phase Diagrams Chemistry / However, this is not the case for sodium sulfate above 30ºc where the solubility then begins to decrease.. The solubility of many substances increases with increasing temperature. The solubility of a gas in a liquid always increases as the temperature rises. Most of the impurities will. 7 90 g of sodium nitrate are added to 100 g of water at 0 deg c. Not all alloys • cu and ni show very different physical properties in their pure states, and the a phase provides a.

What relationship exists between solubility and temperature for most of the substances shown? Ninety grams of nano3 is added to 100 g of h2o at o'c. A solubility curve is a graph of the solubility of a solute in grams per 100 grams of water (shown on the y axis) at various temperatures in degrees celsius (shown on the x axis). Gases are one of the states of matter, either compressed very tightly or expanded to fill a large space. What relationship exists between solubility and temperature for most substances?

Water As The Reaction Medium In Organic Chemistry From Our Worst Enemy To Our Best Friend Chemical Science Rsc Publishing Doi 10 1039 D0sc06000c from pubs.rsc.org What relationship exists between solubility and temperature for the only gas, so2, on the graph? What is the relation between pressure and temperature? How many grams of so2 will bubble out of solution if you melt the ice and raise the temperature of the water to 80. Gases are one of the states of matter, either compressed very tightly or expanded to fill a large space. What explains why solids become more soluble as temperature increases and why gases become less. A saturated solution of kclo3 was made with 300 g of h2o at 34 °c. • exist over the whole composition dissolve more in crystal structure of the higher valence metal than vice versa. Use of instead of is often inconvenient because it is usually the state of the system that we are interested in.

### However, this is not the case for sodium sulfate above 30ºc where the.

7 90 g of sodium nitrate are added to 100 g of water at 0 deg c. For example, sugar dissolves better in hot tea than cold tea. With constant stirring, to what temperature must the solution be raised to produce a saturated solution with no solid nano3. But there are lot of substances that break this rule. How much kclo3 could be recovered by evaporating the solution to dryness? If temperature increases then the solubility also increases. What term is given to a substance that can dissolve in a particular liquid? Temperature is always mentioned along with solubility because solubility of a substance is directly proportional to the temperature. Gases are less soluble at higher temperature, illustrating an indirect relationship 3. What is the relationship between pressure and temperature? The relationship between solubility and temperature can be expressed by a solubility curve. What relationship exists between solubility and temperature for most of the sub stances shown? Ninety grams of nano3 is added to 100 g of h2o at o'c.

Since solubility tables are always in molality, to go from the molality to molarity i would need the density of the solution. The relationship between temperature and solubility will vary depending on the solute and solvent in question. What relationship exists between solubility and temperature for most of the substances shown? A solubility curve is a graph of the solubility of a solute in grams per 100 grams of water (shown on the y axis) at various temperatures in degrees celsius (shown on the x axis). Solubility often depends on temperature

Solubility Wikipedia from upload.wikimedia.org How much kclo3 could be recovered by evaporating the solution to dryness? Many salts show a large increase in solubility with temperature. The solubility of a substance in water decreases as the temperature rises, especially for ionic solids. When you add a solute to a solvent, the kinetic energy of the solvent molecules thus, increasing the temperature increases the solubilities of substances. What relationship exists between solubility and temperature for the ionic substances shown? Since solubility tables are always in molality, to go from the molality to molarity i would need the density of the solution. • exist over the whole composition dissolve more in crystal structure of the higher valence metal than vice versa. Not all alloys • cu and ni show very different physical properties in their pure states, and the a phase provides a.

### What is the equation for the relationship between solubility and.

What is the relation between pressure and temperature? What is the equation for the relationship between solubility and. Most of the time the solubility of a solid will increase with an increase in temperature. The temperature at which melting occurs is the melting point ( mp ) of the substance. What relationship exists between solubility and temperature for most of the substances shown? Grams of water at 63.0°c? As the particles move faster and faster, they begin to break the attractive forces between each other and most substances go through the logical progression from solid to liquid to gas as they're heated — or. Ionic compounds with a high lattice energy will be very soluble. A saturated solution of kclo3 was made with 300 g of h2o at 34 °c. Gases are one of the states of matter, either compressed very tightly or expanded to fill a large space. What relationship exists between solubility and temperature for most substances? Recall the relationship between solubility and temperature. Complete the following table showing the steps in a procedure to determine how does the solubility of solid substances change as the temperature of the solvent increases?

7 90 g of sodium nitrate are added to 100 g of water at 0 deg c. The molecules in a substance have a range of kinetic energies because they don't all move at the same speed. What relationship exists between solubility and temperature for most of the substances shown? What relationship exists between solubility and temperature for the ionic substances shown? A solubility curve is a graph of the solubility of a solute in grams per 100 grams of water (shown on the y axis) at various temperatures in degrees celsius (shown on the x axis).

Solubility And Metastable Zone Width Crystallization from www.mt.com A solubility curve is a graph of the solubility of a solute in grams per 100 grams of water (shown on the y axis) at various temperatures in degrees celsius (shown on the x axis). However, this is not the case for sodium sulfate above 30ºc where the. Temperature is always mentioned along with solubility because solubility of a substance is directly proportional to the temperature. How many grams of sodium nitrate will dissolve in 300. With constant stirring, to what temperature must the solution be raised to produce a saturated solution with no solid remaining? Complete the following table showing the steps in a procedure to determine how does the solubility of solid substances change as the temperature of the solvent increases? What relationship exists between solubility and temperature for the only gas, so2, on the graph? The solubility of a substance in water decreases as the temperature rises, especially for ionic solids.

### Temperature is a measurement of the average kinetic energy of the molecules in an object or a system.

7 90 g of sodium nitrate are added to 100 g of water at 0 deg c. (1988) relationship between solubility and micellization of surfactants: What is the equation for the relationship between solubility and. What is the relation between pressure and temperature? Some substances only dissolve at high temperatures. However, this is not the case for sodium sulfate above 30ºc where the. The solubility of a gas in a liquid always increases as the temperature rises. Graphs below are soluablity curves of some since most compounds are more soluble at higher temperature, lithium carbonate can be purified by heating it in water. Solubility often depends on temperature Ninety grams of nano3 is added to 100 g of h2o at o'c. As the particles move faster and faster, they begin to break the attractive forces between each other and most substances go through the logical progression from solid to liquid to gas as they're heated — or. Most of the time the solubility of a solid will increase with an increase in temperature. • exist over the whole composition dissolve more in crystal structure of the higher valence metal than vice versa.

With constant stirring, to what temperature must the solution be raised to produce a saturated solution with no solid nano3. What relationship exists between solubility and temperature for most of the substances shown? Gases are less soluble at higher temperature, illustrating an indirect relationship 3. Complete the following table showing the steps in a procedure to determine how does the solubility of solid substances change as the temperature of the solvent increases? What is the relationship between pressure and temperature?

Ninety grams of nano3 is added to 100 g of h2o at o'c.  each 9 temperature and gas solubility  unlike most solids, gases become less soluble as the solubility the maximum quantity of the substance, expressed in grams, that. The relationship between temperature and solubility will vary depending on the solute and solvent in question. With constant stirring, to what temperature must the solution be raised to produce a saturated solution with no solid remaining? 1 what relationship exists between solubility and temperature for most of the substances shown?

Grams of ice saturated with so2 at 0. Solid substances dissolved in liquid water, the solubility increases with temperature. (other relationships we work with will typically require an absolute scale, so in these notes we use either the kelvin or rankine scales. Relationship between density and molar mass and pressure. Generally, it can be described by the van't hoff equation.

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Use of instead of is often inconvenient because it is usually the state of the system that we are interested in. The relationship between solubility and temperature can be expressed by a solubility curve. The solubility of most substances improves as temperature rises. What relationship exists between solubility and temperature for most substances? The temperature range of micellization.

The solubility of a gas in a liquid always increases as the temperature rises. What relationship exists between solubility and temperature for the ionic substances shown? What relationship exists between solubility and temperature for most of the substances shown? As a subtance absorbs heat the particles move faster so the average kinetic. How many grams of so2 will bubble out of solution if you melt the ice and raise the temperature of the water to 80.

Ninety grams of nano3 is added to 100 g of h2o at o'c. What relationship exists between solubility and temperature for most of the substances shown? Gases are one of the states of matter, either compressed very tightly or expanded to fill a large space. Which substance's solubility changes most with temperature? Increasing the temperature always decreases the solubility of gases.

1 what relationship exists between solubility and temperature for most of the substances shown? Complete the following table showing the steps in a procedure to determine how does the solubility of solid substances change as the temperature of the solvent increases? Gases are one of the states of matter, either compressed very tightly or expanded to fill a large space. The temperature at which melting occurs is the melting point ( mp ) of the substance. Generally, it can be described by the van't hoff equation.

Temperature is a measurement of the average kinetic energy of the molecules in an object or a system. How much kclo3 could be recovered by evaporating the solution to dryness? Many salts show a large increase in solubility with temperature. Grams of ice saturated with so2 at 0. Graphs below are soluablity curves of some since most compounds are more soluble at higher temperature, lithium carbonate can be purified by heating it in water.

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Generally, it can be described by the van't hoff equation. Grams of ice saturated with so2 at 0. What is the relation between pressure and temperature? The temperature range of micellization. Which of the following states the relationship between temperature and the solubility of a substance in water?

Relationship between density and molar mass and pressure.

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When you add a solute to a solvent, the kinetic energy of the solvent molecules thus, increasing the temperature increases the solubilities of substances.

7 90 g of sodium nitrate are added to 100 g of water at 0 deg c.

With constant stirring, to what temperature must the solution be raised to produce a saturated solution with no solid remaining?

Grams of ice saturated with so2 at 0.

What relationship exists between solubility and temperature for most of the substances shown?

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Ionic compounds with a high lattice energy will be very soluble.

What relationship exists between solubility and temperature for most of the substances shown?

Most of the time the solubility of a solid will increase with an increase in temperature.

 each 9 temperature and gas solubility  unlike most solids, gases become less soluble as the solubility the maximum quantity of the substance, expressed in grams, that.

Use of instead of is often inconvenient because it is usually the state of the system that we are interested in.

Which substance's solubility changes most with temperature?

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Most of the time the solubility of a solid will increase with an increase in temperature.

What relationship exists between solubility and temperature for most of the substances shown?

The relationship between temperature and solubility will vary depending on the solute and solvent in question.

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Many technical terms relating the solubility of surfactants with their aggregation as micelles are reviewed in order to derive a consistent concept of cite this paper as:

Gases tend to naturally have high entropy or kinetic energy than solid substances so the same still applies.

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What relationship exists between solubility and temperature for most of the sub stances shown?

Use of instead of is often inconvenient because it is usually the state of the system that we are interested in.

What relationship exists between solubility and temperature for most of the substances shown?