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# Emergence of function from coordinated cells in a tissue

Contributed by Stephen Smale, December 22, 2016 (sent for review November 4, 2016; reviewed by Sayan Mukherjee and Jean-Jacques Slotine)

## Significance

A basic problem in biology is understanding how information from a single genome gives rise to function in a mature multicellular tissue. Genome dynamics stabilize to give rise to a protein distribution in a given cell type, which in turn gives rise to the identity of a cell. We build a highly idealized mathematical foundation that combines the genome (within cell) and the diffusion (between cell) dynamical forces. The trade-off between these forces gives rise to the emergence of function. We define emergence as the coordinated effect of individual components that establishes an objective not possible for an individual component. Our setting of emergence may further our understanding of normal tissue function and dysfunctional states such as cancer.

## Abstract

This work presents a mathematical study of tissue dynamics. We combine within-cell genome dynamics and diffusion between cells, so that the synthesis of the two gives rise to the emergence of function, akin to establishing “tissue homeostasis.” We introduce two concepts, monotonicity and a weak version of hardwiring. These together are sufficient for global convergence of the tissue dynamics.

Is there a basis for emergence of tissue-specific function? Emergence in this work is defined as the coordinated effect of individual components that establishes an objective not possible for an individual component. Our components are cells with their proteins, and the objective is the function of a tissue. Here tissue is a set of cells of the same cell type located together as exemplified by an organ in the body. In vertebrates, consider the liver, functioning to detoxify and ensure an appropriate composition of blood, and the skeletal muscle, functioning to contract and generate force. In each of these tissues millions of individual cells contribute to emergence of function according to their cell type. The main elements of emergence that we consider are first, the protein distribution in a given cell type and second, the cellular architecture of the tissue, a 3D structure with “diffusion” of molecules between cells. We build a mathematical model for emergence of function, from a large accumulation of data. We also use our previous work on cell dynamics (genome dynamics) and the work of Alan Turing on diffusion (1).

Underlying our setting are widely believed biological hypotheses: (*i*) Cells within a tissue (i.e., the same cell type) have the same dynamics and the same distribution of proteins at equilibrium and (*ii*) the function of a cell corresponds to the proteins of that cell. For reasons that will be discussed, we call the property in *i* “hardwiring” of the tissue (2, 3). Convergence of the tissue dynamics to such an equilibrium naturally takes on importance, for its role in maintenance of tissue function (4). Even a local stability of the (hardwiring) equilibrium, i.e., its robustness, gives some validity to our model in biology. Our main theorem (*Theorem 5*) establishes that monotonicity, a property that we introduce here, implies global convergence of the tissue dynamics to the equilibrium, where all cells have the same protein distribution. This gives a biological justification for our framework and a model for “emergence of function,” as well as suggestions for studying the passage from emergence to morphogenesis. On the other hand one could see the emergence described here as a final stage of morphogenesis, completing a cycle.

Our model could give some support to obtaining more insights. Further questions, where quantitative support is expected, are also suggested: (*i*) To what extent is there a common equilibrium of proteins in each cell in a tissue? (*ii*) How do cells in a tissue cooperate to give rise to function? And (*iii*) how do we measure the diffusion between the cells?

## 1. Simple Example

Here we model two cells, separated by a membrane, that each have a same single protein. Consider the following system,

The equilibrium for the above system is obtained by solving the system derived from Eq. **2** by setting the right-hand sides equal to zero. This is a linear system in two equations and two variables and we obtain

It is not hard to see from Eqs. **3a** and **3b** that if *2. The Genome Dynamics of One Cell* and *5. Turing’s Paper on Morphogenesis*) for the system in Eq. **2** at equilibrium and finite

Because the eigenvalues are real negative, this pair **3** is a stable equilibrium.

### Success of Emergence.

The magnitude of **3** measures the departure from the “emergence” as

If **3**, the solution is **4**, for any finite **2** is not emergent (for any finite

**Remark 1**: Here the

This is an example of linear dynamics of one protein and one cell with stability. These dynamics although linear are also a good approximation of the general stable dynamics in a neighborhood of the equilibria. Moreover, the global dynamics of the basin of the stable equilibrium are qualitatively equivalent to the linear example.

## 2. The Genome Dynamics of One Cell

We use the setting of our paper on genome dynamics (2). For a single-cell state space

Generally recall the notion of stable equilibrium

The dynamics on the basin are “linear” provided that the

**Definition 1: Monotonicity condition**: Suppose we have dynamics

**Proposition 1.** *For any dynamics* *on* *the monotonicity condition for* *implies that* *is a decreasing function of* *for all nontrivial solutions* *in* *where* *is defined for all*

**Proof**: Suppose Eq. **5** is true. Note that

The quantity at the end is negative by Eq. **10**, the monotonicity condition. QED.

One could call the *Proposition 1* a “monotonic basin” for the dynamics. Under these conditions

Thus, monotonicity on *5. Turing’s Paper on Morphogenesis*).

**Example 1**: Let

Let us return to the biological setting. Single-cell dynamics are those of dynamics on a basin

We now examine explicitly the conditions for monotonicity in the linear case of one cell with two proteins. This case can be represented by the system

Then *i*) The trace, *ii*) the determinant,

To derive the conditions for monotonicity, consider the quadratic form associated with

In summary the stability condition is *Proposition 1* we can prove the following.

**Proposition 2**. *For linear dynamics on* *monotonicity implies stability*.

Fig. 3 shows an example of monotonicity and stability conditions in the *5. Turing’s Paper on Morphogenesis*.

### Hardwiring.

The genes present in the human genome are the same in all cell types and all individuals. Now we describe a property of a family of cells, which we called hardwiring (2), motivated by the universality above. Our network in ref. 2 puts an oriented edge (between two nodes), between two genes,

### Definition of Weak Hardwiring.

Thus, the family is hardwired provided that the dynamics of each cell in the family are the same; in particular, the equilibrium of each cell is the same. That is, the protein distribution at the equilibrium of each cell is the same. If the last property is true, then we say that the family satisfies “weak hardwiring.” The idea of the weak hardwiring concept is that in a single cell type all cells have the same equilibrium distribution of proteins (2). This helps justify the identification of a tissue with its protein distribution.

## 3. Cellular Dynamics and Their Architecture

We define a graph *3. Cellular Dynamics and Their Architecture*. This model applies more literally to diffusion in the case of small molecules.

### Definition of a State.

A state associated to the graph is a set of protein levels *i*th cell. Thus, a state *i*th cell; i.e., *5. Turing’s Paper on Morphogenesis*).

### The Laplacian Matrix.

Let *i*th element of the diagonal defined by

The diffusion dynamics defined by the cellular architecture may be written as follows:

Note that Eq. **12** is a linear system of ordinary differential equations.

**Remark 2**: Harmonic functions are exactly a set of

Note that our definition applies not just to a single protein, but also to an

**Proposition 3**. *The system is globally stable with equilibrium set, the harmonic functions*.

**Proof**: **7** is denoted by

Then

For the

## 4. Dynamics of a Tissue

We use both the notations *6. Lapse of Emergence*.

### Genome Dynamics for *m* Cells.

*m*

For a single cell say

For the case of cells of a tissue *3. Cellular Dynamics and Their Architecture* extended to

Now we take the product of the dynamics over all of the cells at once to get

Eq. **9** is rephrasing Eq. **8**. This is the genome dynamics of the tissue. Thus, this tissue has genome dynamics and separates into individual cell dynamics

### Extension of Monotonicity to the Genome Dynamics of the Tissue.

Extension of the definition of monotonicity to many cells is given by

Here *j*th protein in the *i*th cell.

**Example 2**:

However, we are not assuming the linearity of the dynamics. We cannot get even a good model of robust stability of equilibria in the linear setting. We cannot model dynamics with two separate equilibria.

### Diffusion Dynamics for *n* Proteins.

*n*

Recall in *3. Cellular Dynamics and Their Architecture*, the diffusion dynamics between cells in a tissue for a protein distribution

Here

### The Basin Hypothesis.

Cells described by **11** below make sense and are called the basin hypothesis.

Following the spirit of Turing’s paper, we may combine two dynamics (genome dynamics within the cell and diffusion dynamics between cells) into a system (Eq. **11**) that is the object of the study in this paper:

We emphasize that differential Eq. **11** is not necessarily linear in contrast to Turing.

The main theorem of this paper is as follows.

**Theorem 5.** *The dynamical system* *of a tissue* (Eq. **11**) *is globally stable with equilibrium* *provided the basin hypothesis is satisfied and* *satisfies monotonicity*.

**Lemma 1.** *If* *is monotone relative to* *then* *is monotone*.

**Proof of Lemma 1**: Lemma 1 is true if *4. Dynamics of a Tissue*). Therefore,

By *Proposition 1* applied to *Lemma 1* we obtain the global stability of equilibrium *Theorem 5*.

We name the property of *Theorem 5* emergence. *Theorem 5* establishes that monotonicity implies global convergence of the tissue dynamics to the equilibrium; that is, all cells have the same protein distribution, in a strong stable sense (“robustness”). This gives a biological justification for the concept of weak hardwiring in a tissue. Thus, we give a model for emergence of function.

Ours is not the first paper to address the issue of convergence after Turing’s paper. In particular, Wang and Slotine (14) have conditions designed to show convergence in systems resembling ours. Our work in *Proposition 1* and *Theorem 5* deals with a wider class of spaces as our basins and their products correspond to dynamics of protein levels. Also we do not use a derivative of

**Remark 3:** Recall the linear case

## 5. Turing’s Paper on Morphogenesis

The work of Alan Turing plays an important role in our paper. The main differential Eq. **11** owes much to ref. 1. There are some important differences. First, we use nonlinearity for the cell dynamics in contrast to the Turing linear setting. Nonlinearity allows us to address issues of stability, where the second derivative plays a crucial role and we are able to use associated domains of the cell dynamics more in accord with the biology. On the other hand, Turing developed his work in a partial differential equations framework, with reaction diffusion equations that reflect a continuum perspective of the nature of cells. That leads to some applications in morphogenesis, such as patterning in Zebra stripes (18, 19). Our own perspective differs. We feel that some of the basic features of morphogenesis must deal with few cells (embryogenesis, cell differentiation). The recent work of Chua (9) also develops Turing’s contributions in a different direction from our work.

Turing found an important example of the system of the same type we used in *4. Dynamics of a Tissue*. The example shows how a system that is stable without diffusion becomes unstable in the presence of diffusion. Turing was motivated to understand morphogenesis with this example of instability. The example consists of two cells and two proteins. The variables

The two cells are identical in this example and we can describe the cell dynamics as **12** into our form,

### Genome Dynamics of the Turing Example.

We now show that a key phenomenon of this example is the failure of the monotonicity condition. That is necessary to give rise to instability (morphogenesis).

Let us then study the monotonicity of *2. The Genome Dynamics of One Cell* as well as perform a 2D analysis for the Turing example. First we construct matrix

The trace

### Diffusion Dynamics.

The diffusion dynamics in the system in Eq. **12** are expressed by the terms *3. Cellular Dynamics and Their Architecture*, their eigenvalues are nonpositive.

### Full Dynamics.

Combining genome dynamics and diffusion dynamics gives the Turing example of the system in Eq. **19**. The linear part of the system in Eq. **19** is

The system in Eq. **19** has a unique equilibrium that is obtained by solving the right-hand side of the equation set equal to zero. Eigenvalues of

Wang and Slotine (20) examine a condition on the dynamics that depends on the diffusion coupling whereas our monotonicity depends only on the dynamics in each cell and not on the diffusion. But each condition plays a main role in the convergence.

Smale (11) examined similar equations with nonlinear cell dynamics. He considered each of two cells as having a global stable equilibrium, and therefore the cells were “dead” in an abstract mathematical sense. But upon coupling the two cells by diffusion, he proved that the resulting system has a global periodic attractor, and hence the cells become “alive.” Toward this end Smale’s work was a mathematical model similar to the Turing two-cell example but with dynamics of each cell not linear, leading to the model*k*th cell and the second term describes the diffusion processes between cells. The principal case considered by Smale (11) is **13** is precisely a form of our main equations. Again the phenomenon depends on the failure of monotonicity.

### (Easy) Conjecture 1.

*Generically monotonicity of a linear system in Euclidean space is equivalent to all eigenvalues negative* (*and real*).

## 6. Lapse of Emergence

Here we discuss an avenue to study the departure from emergence, using our setting. Consider the Jacobian of **11**) at the equilibrium *i*th cell and

Now look at

This paper can be used to examine the end of emergence in terms of cell division (symmetric or asymmetric or cancer) (21, 22).

## Acknowledgments

We thank Lindsey Muir, Scott Ronquist, and Thomas Ried for helpful discussions and James Gimlett and Srikanta Kumar at Defense Advanced Research Projects Agency for support and encouragement.

## Footnotes

↵

^{1}I.R. and S.S. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: smale{at}math.berkeley.edu or indikar{at}umich.edu.

Author contributions: I.R. and S.S. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.

Reviewers: S.M., Duke University; and J.-J.S., Massachusetts Institute of Technology.

The authors declare no conflict of interest.

Freely available online through the PNAS open access option.

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*Mathematical Biology I: An Introduction*(Springer, New York), Interdisciplinary Applied Mathematics, Vol 17. - ↵.
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