Coupled Systems, Hierarchy and Emergence

Naturally, coupled systems are generally nonlinear. That is always the case if two compounds form a special reaction pathway that is not possible if only a single compound is present. If two molecules of two different compounds interact, then the reaction is bilinear or bimolecular, which is a nonlinearity. If two molecules (or two photons) of the same kind are able to reach a state that is not accessible by a single molecule (or photon) then the outcome is typically in a nonlinear dependence on the input. One prominent example of nonlinear optics in physics is the two photon absorption where the absorbing state is reached by interaction of two photons with the ground state within a certain time interval. If the photon density is too low for that to happen then the output is zero, but when the photon density increases the probability for the two photon absorption increases with the square of the photon density. Therefore two (or more) photons are needed to activate a state transition. Only spontaneous population and depopulation of certain states, which is not the typical situation for characteristic biochemical reactions, are truly linear. If several molecules of one or several compounds have to interact within a certain time interval, then the reaction scheme is nonlinear and characterized by the typical mathematical problems and challenges of nonlinear systems.

Photosynthesis is a truly nonlinear reaction as at least eight photons have to be absorbed by two different photosystems to split two molecules of water and release one molecule of oxygen. Photon absorption drives an electron transfer in photosynthesis. However, the involved molecules are reduced and/or oxidized by more than one electron and the coupled proton transfer again forms ATP from ADP and phosphate in a nonlinear process. Biochemistry is truly a hierarchy of nonlinear processes.

The “cycles” of nonlinearities that form the overall, hierarchical structure also include loss processes. After photon absorption excitation- energy can be lost and the following electron transfer processes are likewise restricted by loss processes that limit the production of one molecule of oxygen with a demand of at least 11-12 photons. Other sources report 60 photons per molecule glucose (Hader, 1999; Campbell and Reece, 2009) which would equal 10 photons per molecule glucose according to the basic equation of photosynthesis understood as the light-induced chemical reaction of water with carbon dioxide to glucose:

The energetic stoichiometry of light and dark reactions in photosynthesis are again discussed in chap. 4.4.1 as well as in the literature (Hader, 1999). This also features discussions of coupled reaction schemes like the proton assisted electron transfer (Renger, 2008, 2012; Renger and Ludwig, 2011). This question shall therefore not be discussed here in more detail. The basic principles of the photosynthetic light reaction are presented in chapter two.

Here we primarily intend to elucidate the highly nonlinear character of photosynthesis. Indeed, the nonlinearity of photosynthesis goes far beyond this discussion. If one regards the hierarchy of the spatiotemporal order of a plant as an overall reaction system, one could ask how many photons are involved in the construction of a new leaf. Analyzing the biomass of a dried leaf, which is strongly dependent on the organism, we might look at, say, 100 mg and find that the fixation of an order of 1021 carbon molecules was necessary with a corresponding nonlinear response of a new “leaf’ to more than 1022 absorbed photons. That means absorption of 1022 photons finally leads to the spontaneous appearance of a single “leaf’. Of course these photons have to be absorbed within a certain time interval. If illumination stays under a certain threshold, nothing happens, but if bright sunlight, sufficient day length and adequate temperature trigger the mechanisms correctly in spring, leaves might appear proportional to “packages” of 1022 absorbed photons. In this sense, the process might seem to be a linear response, but it is surely not and stops completely after a short growth period when new priorities like the production and storage of biomass take over in summer.

Even if we understand this reaction as a subsequent construction which can be analyzed step by step, it might still be a matter for discussion whether this reductionism leads to the loss of information and prevents our overall understanding of the growth of a plant (Heisenberg, 1986). After all, we have the appearance of one single leaf after the absorption of 1022 photons if we work as a pure phycisist who did not learn the details of biochemistry and does not know anything from gene activation and proteomics.

Our first identified nonlinear system (the water splitting and oxygen evolution) forms the trigger or the “input” for processes that are highly nonlinear themselves since they require several molecules of glucose, ATP or NADH to drive the production of one single further unit like for example a whole cell. We have a complicated spatiotemporal network of nonlinear systems that are coupled to nonlinear networks on the next hierarchically higher “level”. In this way, the complexity of the overall system, the plant, arises. However, if only bottom up processes from the molecular interaction on the single molecule level are taken into account, then reductionism will fail to explain the details of the plant’s morphology and lifespan.

Therefore it might still be wondered, like Heisenberg did in his book Der Teil und das Ganze (Heisenberg, 1986), whether we can expect that a possible picture of a plant as an organism understood in full detail will still use the language of physics or whether this picture will require that we formulate its propositions with novel approaches. Of course, taking the view of a physicist, the formalism that enables a scientist to understand an organism in detail and therefore enables the possibility to simu?late the response of the system to a parameter change will be understood as a novel formalism. Therefore, Heisenberg’s skepticism might not necessarily lead to the termination of actual scientific approaches. It is more likely that in the future science will overcome actual limitations as it has always done in the past: by inventing new approaches, new languages and more computational power. Current novel, highly-funded activities in the field of synthetic biology give rise to the hope that at least the composition of a single working cell from its basic chemical compounds will soon be possible.

Outputs of highly nonlinear reaction schemes function as substrate for highly nonlinear processes - on a hierarchic order of the scaling. In fact, interaction partners, cofactors, substrates, a series of several molecules or whole reaction networks, and even time can be a nonlinearity or a nonlinear scaling factor if there exists a feedback parameter that influences the dynamics in such way that it is no longer dependent on the substrate in a purely linear manner. Mathematical and physical representations of quite simple nonlinear systems show that the dynamics of such systems can change extremely if one parameter only changes slightly. This is called “chaotic behavior” or simply “chaos”.

In contrast to the fact that nonlinear systems typically respond with chaotic behaviour, such coupled nonlinear structures can also be stable. Stability can arise from chaos. The chaotic behaviour of simple reaction schemes and the complexity that arises as well as the simplicity that arises from a complex pattern were topics in a broad series of literature that flourished 15-20 years ago (Gell-Mann, 1994; Cohen and Stewart, 1997; Wolfram, 2002). These excellent research books show how simple rules lead to complex structures and/or mathematically analyze chaos in deep detail using a scientific procedure.

At this point we might go a step backwards and focus on our initially introduced question how complex phenomena can arise from simple processes and how a hierarchical organization might deliver patterns with novel properties. It helps also to understand principles of selfsimilarity and macroscopic structuring arising from simple rules for single (chemical) reactions. Stephen Wolfram's book A New Kind of Science deals with such an approach to motivate rise and decay of complex macroscopic patterns from simple rules on the microscale.

He works with so called “cellular automata” that define rules how to color a square in dependency of neighbouring squares. Programming such rules he found that the computer starts to draw interesting patterns and complex networks that arise even if the underlying rule is the most possible simple one working with black and white squares only.

Stephen Wolfram’s underlying aproach is clearly bottom up. The concept of cellular automata explains complex patterns and their selfsimilarity by basic rules that the smallest entities in the concept have to follow. From the simple rule that determines in which color neighbouring squares on a white sheet of paper have to be imaged he can principally generate any desired pattern while discussing complexity, spatial and temporal hierarchies in the structures and the phenomenon of emergence.

Figure 1 is a computation of a simple rule, called “rule 30” by Wolfram with the commercial program Mathematica® also invented by Wolfram and distributed by Wolfram research. Wolfram has invented the concept of cellular automata that propose certain rules for the generation of binary two dimensional (or also more complicated) patterns of elementary cells only from the information content of neighbouring cells. Some selected rules as published in (Wolfram, 2002) are shown in Figure 2.

Mathematica® simulation according to “rule 30” (see Wolfram, 2002 and Figure 2) with 400 lines/iteration cycles. The image is cut asymmetrically at the left and right side

Figure 1. Mathematica® simulation according to “rule 30” (see Wolfram, 2002 and Figure 2) with 400 lines/iteration cycles. The image is cut asymmetrically at the left and right side.

Cellular automata generate patterns starting with a single black elementary cell in the middle of the first line on a white sheet of paper. The rule delivers the information how the neighbouring elementary cells in the next line have to be colored. The output of the rules indicated in

Figure 2 is shown in Figure 3. Some of these rules generate quite boring patterns like rule 222 which forms a black pyramid and rule 250 forming a chess board pattern (see Figure 2 and Figure 3). However, there exist rules that generate complicated patterns with typical properties of selfsimilarity. For example rule 30 (see Figure 1, Figure 2 and Figure 3) iteratively generates white triangles standing on the top with varying size (albeit with a limited size distribution).

Typical rules for cellular automata according to Wolfram (Wolfram, 2002). Image reproduced with permission

Figure 2. Typical rules for cellular automata according to Wolfram (Wolfram, 2002). Image reproduced with permission.

Output of the rules shown in Figure 2

Figure 3. Output of the rules shown in Figure 2.

Interestingly, some cellular automata also typically show properties of “growth” from a single black elementary cell to a complex structure as shown in Figure 1 that is persistent with constant complexity for a certain time interval (or forever). Indeed some of the rules, especially when introducing several colors, show an apparent rising complexity from a rather simple basic structure to a pattern that then decays again and leads to a constant configuration that no longer changes line by line.

When I was younger these cellular automata fascinated me and I thought that they might be the only necessary concept to explain any biological pattern, even the birth and death of life, the diversity of its forms and general emergent phenomena like consciousness. Wolfram understands the unforeseen growth of novel structures on the patterns generated by cellular automata as emergence. And he refers to his automata and their possibility to derive complexity from a few simple rules directly to biology. In his index, the entry for “emergence, general concept” points to page 3 (Wolfram, 2002) where the word is not found a single time. Instead Wolfram states:

“It could have been, after all, that in the natural world we would mostly see forms like squares and circles that we consider simple. But in fact one of the most striking features of the natural world is that across a vast range of physical, biological and other systems we are continually confronted with what seems to be immense complexity. And indeed throughout most of history it has been taken almost for granted that such complexity - being so vastly greater than in the works of humans - could only be the work of a supernatural being. But my discovery that many very simple programs produce great complexity immediately suggests a rather different explanation. For all it takes is that systems in nature operate like typical programs and then it follows that their behavior will often be complex. And the reason that such complexity is not usually seen in human artifacts is just that in building these we tend in effect to use programs that are specially chosen to give only behavior simple enough for us to be able to see that it will achieve the purposes we want” (Wolfram, 2002, page 2f.).

Indeed, Wolfram here performs a stroke of genius in his proposition that the natural behavior of a machine following simple rules is the production of a complex output and that we therefore do not have to answer the question why nature is complex. Rather, complexity is natural and what we need to ask is why mankind thinks in such simple manner that everyone picks the few algorithms from the broad range of possible rules that Wolfram proposes in order to generate simple enough output. This output is not prone to complexity, rather it serves to avoid the possibility that the output might stress the architect’s mind. This finding is highly plausible and there should not be any doubt or any restriction to the work of Wolfram. Nonetheless, some questions remain unanswered. These questions can be formulated as top down or bottom up, a bit like the general concept of thinking that should be kept in mind while reading this book. From bottom up we might ask: why do we have chemical reactions of higher order introducing such strong nonlinearities (such as the nonlinear optics of photosynthesis) assuming that a more simple pattern generator is already capable of simulating any natural behavior? Why does nature behave nonlinearily? When does it produce chaotic output? The output of Wolfram’s cellular automata shows high complexity and self-similarity - however, these automata do not behave in a chaotic way. Or do they? If we change a rule “slightly” the output might change completely. If we shift a black square the output does not change at all. However, it does if we shift the position of the lines next to each other.

Chaos should generally be defined as the fact that the deviation of initial conditions leads to an inherent local inpredictability that rises exponentially over time. Even assuming chaos as deterministic, limited computational power always limits the predictability of an output that arises later in time. Does chaos arise from nonlinearities and is chaos theory a special point of view on nature that can be fully described by imaging all processes on a computer program? It might be possible and in the following we will try to elucidate and answer this question. The concept of computational irreducibility is the final reason for Wolfram to assume that computational models can sufficiently describe nature and especially the reason why it is absolutely necessary to take such computational models into account. Irreducibility is the reason for the generation of intrinsic randomness, which means that his cellular automata inherently generate randomness and he even proposes that neither chaos theory (which is deterministic) nor stochastic perturbations nor quantum theory are necessary for randomness.

So there might be some “features” of natural systems already on the molecular scale that are not implemented in cellular automata. From the other perspective, a top down argument might exist that calls into question whether cellular automata are sufficient to describe any observed natural entity and cover the full range of all existing emergent phenomena. For example, the existence of emotions, or, to put it in a wider sense, consciousness, might be an emergent phenomenon that does not necessarily arise from a computational approach that aims to explain the dynamics of the universe on the scale of mankind employing cellular automata. We cannot answer this question at this point. It is an interesting phenomenon that living systems develop consciousness.

Some authors state that already the unpredictable behavior of irreducible algorithms is sufficient to explain consciousness. They take an external point of view on the program to argue that it has a free will; free will is generally considered to be the most powerful argument against full determinism in nature. However, it might be a phenomenon that emerges from basic simple deterministic rules. At least Wolfram seems to assume so when he states:

“One might have thought that with all their successes over the past few centuries the existing sciences would long ago have managed to address the issue of complexity. But in fact they have not. And indeed for the most part they have specifically defined their scope in order to avoid direct contact with it. For while their basic idea of describing behavior in terms of mathematical equations works well in cases like planetary motion where the behavior is fairly simple, it almost inevitably fails whenever the behavior is more complex. And more or less the same is true of descriptions based on ideas like natural selection in biology. But by thinking in terms of programs the new kind of science that I develop in this book is for the first time able to make meaningful statements about even immensely complex behavior” (Wolfram, 2002).

This is certainly a self-assured stance.

Luckily, since it is still open to question whether plants develop consciousness, we do not have to determine whether consciousness can arise from a computational approach to describing nature for the purposes of this book. The book rather wants to elucidate the complex interplay of nonlinear systems with feedback loops on a spatiotemporal hierarchy of scales based on the example of ROS networking in plants.

Life couples nonlinearities to each other. Evolution means stability in time due to the adaption to a given environment under competition for resources and evolution describes change if the environment changes or if the genetic diversity changes. Nature selects the structures that are observable and therefore there exists a natural selection mechanism that prints the few stable structures emerging by cellular automata or chaotic nonlinear systems into reality. These aspects of nonlinear feedback in evolution and possible meanings for top down signalling are elucidated in chapter five. Thereby we will show that pure Darwinism is sufficient to explain both bottom up as well as the “type” of top down phenomena that we investigate in this book. There is absolutely no need to give space for an intelligent designer of any kind. In this way, we are in full agreement with Wolfram and most recent researchers on the topic.

We do not know whether the emergence of emotions, consciousness, or the fundamental and rather complicated question of the existence of a free will is somehow accessible by concepts like cellular automata based on rate equations to describe their dynamics. Therefore we choose this theoretical approach to describe the most complex (biological) system that is typically described by rate equations in the literature, but that does not bring us into the complication of accounting for striking behavior that is dominated by an emerging ununderstandable effect like consciousness, namely plants.

A top down control feedback in plants is for example represented by the feedback given by a certain concentration of ROS generated by macroscopic events like stress. Stress is a macroscopic event that can even be activated based on free will running on the consciousness, but it does not necessarily need to. It can just arise from environmental conditions. On the other hand, consciousness emerges from the molecular reactions.

Current books on the market elucidate the novel topics in the field of primary processes of photosynthesis (Renger, 2008, 2012; Renger and Ludwig, 2011), nonphotochemical quenching (Demmig-Adams et al., 2014) and, more specifically, the generation and decay of reactive oxygen species (Del Rio and Puppo, 2009). Some authors state that the actual overview on ROS is missing a generalized link to plant morphology and evolution. Even if it might be impossible to achieve such a link without a certain degree of speculation, it is necessary to deal with the topic in a fundamentally scientific manner in order to avoid the postulation of novel unexplainable and experimentally inaccessible entities such as Rupert Sheldrake’s proposition of a “morphogenetic field” (Sheldrake, 2008) or, much worse, theories of an “intelligent designer” that not only compete with recent scientific findings but even actively reject them and support finally fundamentalists and rabble-rousers. We always pay careful respect to the first, basic property of a good scientific theory: the theory must be falsifiable. Everyone who demands inerrability is surely wrong.

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