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More is different with a vengeance

  • Michael P.H. Stumpf
    Correspondence
    Corresponding author
    Affiliations
    Melbourne Integrative Genomics, University of Melbourne, Australia

    School of BioScience, University of Melbourne, Australia

    School of Mathematics and Statistics, University of Melbourne, Australia
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      In his 1972 landmark paper “More is Different,” Philip W. Anderson established “complexity” as a fundamentally important subject of inquiry. He highlighted the profound limitations of reductionist approaches in understanding nature’s complexity, and he set in motion new lines of investigation that have, among other things, led to systems biology.
      P.W. Anderson’s “More is Different” () is 50 years old. Science has grown and changed over the past five decades, and many of the questions from 50 years ago have been solved, at least in parts. But some of the questions have remained topical. Others, including the question as to how complexity arises, have become more mainstream. In 1972 there was no such thing as complexity science, and even chaos theory – or non-linear dynamics – was in its infancy, though rapidly growing (
      • May R.M.
      Simple mathematical models with very complicated dynamics.
      ).
      Anderson’s paper has been influential in setting an agenda for the future of complexity science. It has also been deeply influential on the outlook of many in the field of complexity science, but also beyond, including statistical physics, mathematical and systems biology. Anderson's writing was almost certainly in parts a response to prevailing discussions in physics about what constitutes fundamental research. Elementary particle physics and cosmology have often been portrayed as dealing with the most fundamental aspects of nature and of our universe. Anderson took umbrage to this point of view, not just here, but elsewhere, too (
      • Anderson P.W.
      Concepts in Solids: Lectures on the Theory of Solids.
      ).
      “More is Different” has made three important and lasting contributions: First, Anderson laid out the case for broken symmetry as an organizing principle to aid understanding of dynamical characteristics of complex systems. Second, he showed that the levels at which different processes occur require our close and careful attention: understanding the dynamics at a given level often requires more than simply being able to characterize the dynamics at lower levels. Finally, Anderson set out forcefully the argument that complexity is fundamentally important in its own right.
      Symmetry is an important organizing principle in physics, just as common descent from a single shared ancestor is an organizing principle across the biosciences (
      • Stadler T.
      • Pybus O.G.
      • Stumpf M.P.H.
      Phylodynamics for cell biologists.
      ). Emmy Noether’s fundamental realization (
      • Kosmann-Schwarzbach Y.
      • Schwarzbach B.E.
      The Noether theorems: Invariance and Conservation Laws in the Twentieth Century. Sources and Studies in the History of Mathematics and the Physical Sciences.
      ), and her eponymous theorem, that symmetries (“different viewpoints from which the system appears the same” []) can be related to fundamental conservation laws—conservation of energy is a result of symmetry with respect to translation of time; conservation of momentum is related to symmetry with respect to translation in space; and angular momentum follows from symmetry with respect to rotations, etc.—has been a powerful tool in physics over the past century. Some symmetries have much less straightforward explanations, but the corresponding conservation laws have been guiding research in theoretical physics for the past century (
      • Kosmann-Schwarzbach Y.
      • Schwarzbach B.E.
      The Noether theorems: Invariance and Conservation Laws in the Twentieth Century. Sources and Studies in the History of Mathematics and the Physical Sciences.
      ).
      But symmetry can be lost (or gained) in ways that are unexpected. Anderson’s own research in condensed matter theory (
      • Anderson P.W.
      Absence of diffusion in certain random lattices.
      ;
      • Anderson P.W.
      Concepts in Solids: Lectures on the Theory of Solids.
      ) was closely related to the phenomenon that is now known as broken symmetry. Broken symmetry refers to a scenario where a system has symmetries different from the symmetries of the underlying dynamics. Some of the terminology can appear counter intuitive but a good way to cut through some of the jargon is simply to remember that water is more symmetrical than ice, see Figure 1A: if we rotate or move the right half of Figure 1A it will look, to all intents and purposes, indistinguishable from the original image. For the icy state of water, this is clearly not the case. Crystals in general have recognizable symmetries but they are less symmetric than a spatially homogeneous arrangement. In the case of ice we would have to rotate the arrangement by 60° to get an identical view of the system. A spatially homogeneous system, such as liquid water looks the same from every direction. In the same sense a sphere is more symmetric than a cube: there are more operations that we can perform on a sphere than on a cube so that it will still look the same to an observer.
      Figure thumbnail gr1
      Figure 1Illustration of broken symmetry
      (A) The icy state of water has lower symmetry than the homogeneous liquid water state.
      (B) In the antiferromagnet neighboring spins are aligned in an anti-parallel fashion. For D>2 dimensions as T is increased from 0 Kelvin, spins are flipped randomly and the system eventually displays no long-range order above some critical temperature, the so-called Néel temperature (
      • Anderson P.W.
      Concepts in Solids: Lectures on the Theory of Solids.
      ), and all blue(red) spins are equally likely to point up (down) as down (up).
      (C) The DNA double helix in all living systems on earth is right-handed; left-handed DNA would be just as likely a priori on biophysical grounds as the observed B DNA (left-hand side of 1(c)). In reality, however, a left-handed form of DNA,m Z-DNA (right-hand side of C) is rare and differs biochemically, biophysically, structurally, and functionally from B-DNA.
      Anderson’s main research, which earned him a share of the 1977 Nobel prize in physics was concerned with the gain and loss of order in condensed matter and the physical consequences thereof (
      • Anderson P.W.
      Absence of diffusion in certain random lattices.
      ). For example, Figure 1B depicts a small part of the anti-ferromagnetic Ising model in 2D (later we will briefly return to lower dimensions, e.g. D = 2, where at non-zero temperature, T > 0K = −273.15 C, there can be no long-range order, and hence no magnetism). Here neighboring spins are aligned in an anti-parallel fashion: all the blue spins point “up” and all the red spins point “down”. The net-magnetization of this system is zero as neighboring magnetic moments (indicated by the arrows) cancel each other out. This system has broken symmetry in a more involved sense. First, because of the crystal structure rotational symmetry is broken. Second, because of the magnetic long-range order the time-reversal symmetry is broken. Importantly, the equations of motion that dictate the dynamics of the spins are more symmetrical than the total state of the system, which is “broken”. This broken symmetry state is only attained once the number of sites, effectively becomes infinite, N (and only approximately for finite sized N). There are important lessons to be learned from this example. The most important one is that understanding the collective dynamics of large spin systems has to start from the macroscopic observation, e.g. that the material is anti-ferromagnetic (for example in transition metals), and not from the microscopic equations of motion. The correct way to study the dynamics of broken symmetry systems is therefore to include the new symmetry explicitly and from the outset; the microscopic dynamics do not suffice.
      Figure 1C shows a different type of broken symmetry. The DNA double helix in natural systems is right-handed; so-called B DNA is by far the most common form of DNA. Left-handed DNA is much rarer, and physiologically Z-DNA (on the right of Figure 1 (c)) has very limited functionality. The question as to why right-handed DNA prevails in life on earth is still not settled. But handedness is also observed at the level of sugars and amino acids. With the exception of Glycine, the only non-chiral natural amino acid, the natural amino acids in biology are all left-handed. Naturally occurring sugars by contrast are all right-handed. We call this uniformity of handedness among a class of naturally occurring molecules homochirality. And it is a clear example of a broken symmetry: chiral molecules and their mirror images do not look the same. The remarkable thing about this observation is that all sugars produced by living organisms have the same chirality; the same is true for amino acids. If we synthesize these molecules in the lab we produce both left-handed and right-handed molecules to the same degree. So life itself was required to break this (so-called parity) symmetry and explain chirality of these biologically important molecules. Chirality of DNA, amino acids and sugars does not follow from the laws of quantum mechanics; and no quantum effect or thermodynamic fluctuations can change this asymmetry. The disconnect between the symmetries of a system and the laws which govern it is a fundamental message Anderson distilled in this paper. He uses this as starting point to discuss the new dynamics that emerge when we start considering systems with different levels of complexity: in general, he posits, loss of symmetry goes hand in hand with greater “complication”. Understanding the parts of a system, he argues, is important and fruitful. But we cannot rely solely on this knowledge, generated by reductionist approaches, to generate or reconstitute the complexity at the higher (or systems) level. Each level deserves care and attention and may require its own dedicated theoretical treatment. We therefore have different theoretical frameworks that are appropriate at different levels, see Figure 2. While quantum mechanics and quantum field theory describe elementary particles, atoms, molecules etc, at some stage the quantum mechanistic perspective does not suffice to explain biological complexity.
      Figure thumbnail gr2
      Figure 2Illustration of levels of complexity in science (many more could be added to this figure)
      Each level depends on the previous, lower-complexity levels, but typically we cannot understand the behavior at one level solely in terms of the underlying dynamics. As a result there is no single universally appropriate theoretical/modeling framework that applies across all levels. For many levels we do have different modeling frameworks that may help us to capture—and hopefully to understand—their dynamics and behavior.
      Anderson acknowledged the power of reductionist approaches: dissecting and characterizing the constituent parts is a crucial step in any attempted to gain understanding of complex real-world systems and their dynamics. But he was also a vehement and relentless critic of strong forms of reductionism, arguing that synthesizing complex systems from knowledge of the components' dynamics or behavior will be futile. Condensed matter physics is more than applied elementary particle physics in the same way that biology is more than applied (bio)chemistry. Identifying where, when and which symmetries are broken becomes an essential first step in the analysis of our system of interest. The strategy required to shed light on complex systems in physics is well developed (
      • Sethna J.P.
      Sethna. Statistical Mechanics: Entropy, Order Parameters, and Complexity.
      ). For example, detailed analysis of the loss of long-range order in phase transitions can yield profound and often unexpected results (for example, that there can be no broken symmetry states in two dimensions).
      In biology – with the possible exception of many problems in biophysics, see e.g. (
      • Morris R.G.
      • Rao M.
      Active morphogenesis of epithelial monolayers.
      ) – the situation is somewhat less well developed. And while we know that life in two spatial dimensions is not possible, concrete mathematical results of a similar nature about symmetry breaking in biological systems (
      • Li R.
      • Bowerman B.
      Symmetry breaking in biology.
      ;
      • Zhang H.T.
      • Hiiragi T.
      Symmetry breaking in the mammalian embryo.
      ), appear to remain elusive. The laws that drive the evolution in time of a complex biological system, say cell-fate decisions, the structures and patterns in a tissue, or the ecological dynamics on an island are not naturally described by Schrödinger’s equation. And we cannot start from first principles (or ab initio) and expect that we will obtain a clear and meaningful description of such systems. Instead we must build models based on domain expertise and empirical knowledge (

      Csete M E and J.C Doyle. Reverse engineering of biological complexity. Science, 295:1664-1669, 10.1126/science.1069981, Mar 2002.

      ;
      • May R.M.
      Uses and abuses of mathematics in biology.
      ;
      • Vittadello S.T.
      • Stumpf M.P.H.
      Open Problems in Mathematical Biology.
      ). That means that currently many modeling frameworks are pursued in parallel, see Figure 2, for many of most exciting problems in mathematical biology (and some of the modeling frameworks mentioned in Figure 2 are almost preposterously amorphous). Anderson would probably be completely unperturbed by this. And arguably this diversity reflects the level of excitement in mathematical biology, where even very fundamental questions are wide open. Anderson was certainly prepared to “pile speculation on speculation” and touched upon questions that remain important in biology to this day. How information is stored, processed, and organized, he suggests needs to be understood in terms of broken symmetry, including periodicity of patterns. Even today, and in spite of vast and extensive experimental and theoretical analyses we still know too little about e.g. cellular and neuronal information processing, despite their obvious fundamental importance in biology.
      At the end of “More is Different” Anderson speaks of the “arrogance of the particle physicist” and “some molecular biologists” () and warns of the dangers this and myopic reductionism pose to our understanding of the natural world. The arrangement of disciplines in Figure 2 is only a rough sketch, and most researchers would feel strongly that one or more additional levels need to be added. I would not dispute this. But the important points remain: (1) at each new level new dynamics and new behavior may appear that cannot be fully understood (or which naturally emerges) from understanding the laws and dynamics and the lower level; and (2) there is separation between levels that means that knowledge at one is no longer fully informative about the next level(s) up. How far this sphere of influence extends is difficult to settle, however; but that it is limited, be it the relevance of quantum-field theory to developmental biology, or the relevance of genetics to the development of rich facets in culture, seems sometimes forgotten. Anderson in “More is Different” encourages us to think about this more clearly.

      Declaration of interests

      Michael P.H. Stumpf is on the Advisory Board of Cell Systems.

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