The cellular automaton interpretation of aging and cancer

Table of contents

Introduction

Main Part

Discussion

Summary

References

Introduction

Cells of multicellular organisms communicate with one another through gap junctions that control the movement of ions and other molecules from the cytoplasm of one cell to the cytoplasm of adjacent cells. Gap junctions are composed of connexin proteins [53]. We define a Gap Junction Bioelectric Network (GJBN) as the totality of gap junction communication among a mass of cells.

Cellular automata are mathematical models of many natural phenomena, including biologic processes such as shell patterns in mollusks [1]. Simple automaton rules can result in complexity, and there is no hierarchy of complexity once a threshold is reached [1, 3]. The threshold for complexity is low and rather easy to reach. More complex rules do not increase complexity. Wolfram cellular automaton #110 is the simplest such rule [1].

Cellular automaton #110 is complex, Turing complete, and capable of universal computation. It is bound by the Principle of Computational Equivalence (PCE), and the Principle of Computational Irreducibility (PCI) [1]. If the GJBN is a complex system, then the PCE holds that #110 is sufficiently complex to model it. The PCI teaches that there are no ‘shortcut’ formulas allowing one to calculate the future state of a complex automaton merely by plugging in a future time (Tfuture). Instead, one must ‘run’ the automaton to see its outcome. In the same way, GJBN modeled by cellular automaton #110 must be ‘run’ to determine the outcome of its computations.

This paper proposes that:

- Computations of the GJBN can be modeled by cellular automata.
- Entropic dysregulation of a complex GJBN results in aging and cancer.

‘Cracking the bioelectric code’ [7] implies there might be set of equations which can model the complex biochemical and bioelectric activities of a multicellular organism [5]. The PCI, however, implies there are no ‘shortcut’ equations able to adequately describe the complex biochemical interrelationships of an organism’s environment, gap junctions, and genome. Instead, a class 4 Wolfram cellular automaton such as #110 is required and sufficient to model these relationships accurately and comprehensively [11, 13, 42, 44, 46, 58].

Symbolize a GJBN modeled by Wolfram cellular automaton #110, as GJBN110. Gap junction conductance is regulated by the : 1. production of connexins; 2. numbers and types of connexin channels; 3. ratio of connexin formation vs. destruction; 4. percent of open connexin channels; 5. voltage gating; 6. dispersion and internalization of connexin channels into the cytoplasm; 7. cell membrane potential (Vmem); 8. phosphorylation; 9.ion concentrations; 9.antibodies [14, 15, 34, 43].

Cellular automaton #110 can emulate any other cellular automaton [1]. If by choosing a specific ‘block’ or ‘neighborhood’ of automaton cell values at time ‘T’, one can determine the subsequent value of a cell at time ‘T+1’, then #110 can emulate any another cellular automaton. Importantly, just as a ‘block’ of #110 cell values can be made to emulate any cellular automaton X, so too can GJBN110 alter and combine its gap junction conductivities to emulate any GJBNX.

Information flow in a cellular automaton depends on the Wolfram class of the automaton. For example, in cellular automaton #0, a class 1 cellular automaton, there is no information flow. In #30, a class 3 random cellular automaton, information flow is uncontrolled, and spreads to every part of the automaton. Information flow in #90 produces a nested pattern. In #110, a class 4 automaton, information flow is controlled and confined to moving and colliding structures.

It is important to note that GJBN110 remains in overriding control even when it is emulating another cellular automaton.

Networks modeled by cellular automata are topologically robust. [8, 20, 27, 37, 64] These networks operate within a ‘homeostatic range’ with a well-defined mean, µ, and a narrow Vmem standard deviation, σ. See Figure 3.

The entire multicellular organism, modeled by #110, is symbolized as a scale-free gap junction bioelectric network by writing GJBN*110, where the asterisk symbolizes all fractal scales of the organism : 1. auto-catalyzing, self-organizing molecular networks which lie at the base of larger-scale structures such as DNA and its transcriptional availability as homochromatin; 2. the mitochondrion and other cell organelles; 3. the cell and its gap junctions, and; 4. tissues, and organ systems. [8, 20, 29]

All of these scales are interlocked, each affecting and being affected by the others. They form a self-organizing, self-repairing, fractal, small-world network [8, 11, 23, 24, 27, 29, 30, 42, 45, 46, 65, 66].

Although communication among organ systems depends on hormones and other chemicals (e.g. acetylcholine), we still consider a multicellular organism as a Gap Junction Bioelectric Network because gap junctions ultimately connect the system at the cellular level. Let ‘X’ represent the Wolfram number of any 1-dimensional cellular automaton. Networks of biochemical reactions in GJBN*X can be modeled as information flow gradients in cellular automaton X. Damage at any scale causes damage at all scales. See Figure 1.

Main Part

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Figure 1

Figure 1 illustrates the scale-free, fractal structure of a multicellular organism. The complexity of the system requires class 4 cellular automaton #110 to model its behavior. The system is bound by the principle of computational irreducibility (PCI) meaning that ‘shortcut’ differential equations cannot comprehensively describe the behavior of the system.

Figure 2 illustrates how gap junction movements of negatively-charged chloride ions (electrons) determine cell membrane potential (Vmem), and Vmem gradients. Morphogens are signaling molecules that can activate various genes and gap junction conductivities depending on the morphogen’s local concentration and its concentration gradient. Vmem gradients model morphogen gradients [16, 55].

Figure 3 illustrates V mem control of cellular behavior [5]. Vmem affects the genome, and, reciprocally, the genome affects Vmem by governing the production and regulation of connexins, morphogens, and many other cell membrane components. Morphogens affect gap junction conductivities—allowing GJBN*110 to emulate any GJBN*X.

The basic rules governing GJBN*110 are simple just as cellular automaton rule #110 is simple, but the outcome of its computations are complex (PCE), and bound by the principle of computational irreducibility (PCI). This means that current models of embryogenesis, morphogenesis, aging, and cancer that are based on differential equations are at best only approximations, and cannot capture all the biochemical and computational complexities of a multicellular organism. Analysis of the complexities of a multicellular organism will require models based on GJBN*110. Inasmuch as simple rules can lead to complex results, development of these computer models may not be as difficult as one might suppose. A common mistake is making the model too complex [11].

GJBN*110 models the spatio-temporal patterns of Vmem potentials and Vmem gradients that define and guide the overall geometry of embryogenesis, morphogenesis, healing, regeneration, and the dynamic equilibrium of morphostasis responsible for structural stability. A youthful morphology depends on maintenance of morphostasis despite continual damage from entropy. Aging may be a consequence of loss of morphostasis (youthful structural stability) due to entropic degradation of GJBN*110.

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Figure 2

Figure 2 illustrates a row of biologic cells connected by gap junctions (short arrows) controlling the flow of chloride ions (e.g., electrons—negative charges) between the cytoplasm of cells. Gray-colored boxes represent cells with intermediate Vmem values depending on their cytoplasmic chloride ion concentration. If negatively-charged chloride ions move to the right, then cells losing cytoplasmic negative chloride ions exhibit hypopolarized membrane potentials with a lower Vmem and a tendency for mitosis, while cells receiving increased cytoplasmic negative charges become hyperpolarized with a higher Vmem and a tendency for differentiation as illustrated in Figure 3 below. A Vmem gradient is established locally. The slope of the gradient (shallow or steep) depends on gap junction conductivities. Because many types of ions can flow through gap junctions, then a multitude of different Vmem gradients can be established. The local concentration of a morphogen activates specific homeobox genes. In a 3-dimensional syncytium of biological cells, a morphogen gradient may not be identical in every direction—the system is anisotropic. Thus, body structures can have different cross-sectional shapes along each dimension.

Vmem↑ Hyperpolarized (quiescence or differentiation)

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Figure 3

Figure 3 illustrates the relationship of Vmem and cellular behavior [5]. If a cell is maximally hyperpolarized, it becomes quiescent. If Vmem is less hyperpolarized, the cell might differentiate into an adult cell. At lower Vmem values, mitosis occurs. At an even more hypopolarized Vmem, the cell might become neoplastic or senescent, until at the lowest levels of hypopolarization, the cell can die. Low Vmem values favor cancer, unbridled mitosis, apoptosis, and perhaps senescent cells (neoplasia protection?). Vmem values probably determine which morphogens are produced by the cell; conversely, morphogens can affect Vmem values.

Spatio-temporal Vmem gradients and morphogen gradients of biologic cells can be modeled by GJBN*110. In Figure 3 above, the dotted line within the smaller bracket indicates the ‘normal’ homeostatic mean Vmem value, µ, of cells. The smaller bracket indicates the ‘normal’ homeostatic range of the standard deviation, σ, of cells when they are controlled by GJBN*110. If the homeostasis of GJBN*110 is dysregulated by entropic damage, then the standard deviation σ of Vmem increases, σ↑ (larger bracket). Cells undergoing excessive, unnecessary, and uncontrolled mitoses may reach the Hayflick limit prematurely and die. Organisms appear to be trapped between cancer and senescence—a trade-off between neoplasia and aging. With loss of GJBN*110 control, and an increase in the standard deviation (σ↑), low Vmem cells can be at risk for cancer. Cellular senescence is a mechanism that may help prevent cancer. If entropic damage results in cells becoming more independent (unicellular), and less under the control of a gap junction bioelectric network, then there is an increase in the standard deviation (σ↑) of Vmem.

Starting from randomness, #110 self-organizes, forming a complex pattern

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Figure 4

Figure 4 illustrates a portion of Wolfram complex cellular automaton #110 [1]. A horizontal row of the automaton models the spatial distribution of scale-free Vmem values and Vmem gradients of biologic cells at time T. Reading vertically down the cellular automaton illustrates the temporal changes in scale-free Vmem values and gradients from T to T+N. Control of connexins and gap junction conductance (e.g. by morphogens) allows GJBN*110 to emulate any GJBN*X. These emulations are dynamic, occur in ‘real time,’ and are epigenetic. Over longer time scales they interact with the genome and the environment, and emulations change as the organism develops, grows, heals, and ages. Morphogen concentrations and gradients can be modeled by GJBN*110 or one of its emulations [46].

Figure 4 illustrates how, starting with an initial random pattern, cellular automaton #110 quickly self-organizes into a complex pattern [8, 20, 48, 64, 65, 66]. Note that if one were to ‘insert’ a new row with a random pattern into any row of #110, the automaton would once again quickly reorganize itself to form a complex pattern. Consequently, if there is entropic damage to GJBN*110, it can also self-repair and reorganize itself [11, 20, 27, 37 ]. See Figures 5a and 5b below.

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Figure 5b

In Figure 5b the major hub and its connections illustrated in 5a have been lost as depicted by the lighter colors. The network has repaired itself by forming new connections that ‘bridge’ the gap created by the loss. The ‘bridges’ indicate repair of the network with a virtual node [6, 8, 11, 13 , 20, 23, 24, 25, 27, 29, 31, 32, 42].

In a cellular automaton, let a darker cell color (Figure 2 above) indicate a hyperpolarized Vmem leading to differentiation, and a lighter cell color indicate a hypopolarized Vmem leading to mitosis. Increasing the number of cell colors (as in a totalistic cellular automaton [1]) produces no new cellular automaton patterns so that if biologic cells have multiple Vmem values, then GJBN*110 can still serve as a model.

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Figure 6

Figure 6 illustrates a totalistic cellular automaton [1] in which the color of the cell is the ‘average’ of the colors of ‘neighborhood’ cells. Note that these cells of varying colors can represent the various Vmem values of biologic cells in a gap junction bioelectric network, as in Figure 2 above. Therefore, a one-dimensional, two-color cellular automaton can model a 3-dimensional, multi-color cellular automaton. This means that a 3-dimensional mass of biologic cells with many Vmem values and many anisotropic morphogen gradients can be modeled by a 1-dimensional 2-color cellular automaton and its emulations. The basic patterns in 1-dimensional cellular automata are representative of all patterns seen in higher dimensions.

Scale-Free Vmem, and Vmem gradients at time T and at time T+N

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S3 S2 S3 S1 S3 S2 S3

Figure 7

Figure 7 illustrates the nested, fractal pattern of cellular automaton #90 (Sierpinski triangle) with self-similarity at all scales. Cellular automaton #110 can emulate cellular automaton #90 by combining several cells such that a ‘block’ (neighborhood) of automaton cells determines the value of a subsequent automaton cell. Similarly, GJBN*110 can emulate GJBN*90, by altering gap junction connectivity such that a ‘block’ of biologic cells determines the subsequent Vmem value of a biologic cell [77]. If morphogens can alter the conductivities of gap junctions, then it follows that as a morphogen diffuses through a mass of cells it could allow GJBN*110 to emulate GJBN*90. Increasing gap junction conductivities could result in a biological cell having a Vmem that is some function of the biological cells in its neighborhood—a criterion for emulation similar to that which occurs in a cellular automaton. The vertical arrows in Figure 7 show growth of light and dark regions as time passes. In this two-color 1-dimensional cellular automaton, white represents mitosis and black represents differentiation (see Figure 3). Depending on the ‘scale’ or nesting of the black (differentiating) cellular automaton cell (and local morphogen gradient), a nested gene from the same scale is ‘called’ or activated. This means that genes for different tissues such as bone, muscle, etc., as well as morphogens are activated at the correct location, scale, and time. Thus, this cellular automaton model can also account for the scale invariance of growth. Genes are nested or scaled in the sense that different parts of the genome ‘program’ are called or activated depending on the scale (as in Figure 7, relative to rest of organism) of the structure being formed [74, 75, 76]. Also note in Figure 7 that black and white cellular automaton cells are mixed at all scales so that mitosis and differentiation also occur together at all scales.

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