Ok, then we'll go with "derived from". Molecular structures are not derived from atomic physics (which are not derived by elementary particles, but defined in terms of these), and we can't even understand cellular dynamics in terms of molecular physics, let alone somehow "derive" any science of living systems from our understanding of molecules. Hell, we have to
assume that gases and liquids and other systems can, in principle, be understood in terms of the dynamics of their constituent elements (i.e., that gases and so forth are made up of molecules and atoms that obey the laws of physics governing the dynamics of atoms and molecules) because even in the Lagrangian or Hamiltonian formulations of classical (analytical) dynamics the equations for tiny amounts of gases require integrals in such high dimensional spaces as to be forever unsolvable even by approximation methods. We rely instead on statistical descriptions that we assume are descriptions of averages of the behaviors of the molecules making up gases and similar systems, and we know that in general these assumptions are wrong but often approximately correct.
Indeed, it is in condensed matter physics, continuum mechanics, and similar levels of analysis in which we find physicists and research in physics that reflects the autonomy (i.e., non-derivability) of the level of analyses that your chart assumes to be non-autonomous (i.e., it assumes higher levels are derived from lower). Even at the level of solid state physics, condensed matter physics, and other areas of physics that require the use of quantum theory (sometimes beyond quantum mechanics) we find the emergence of structure, dynamics, and phenomena that cannot be derived from atomic or elementary particle physics. Certain collective, irreducible states begin to emerge at very low-level analyses.
See e.g.,
Falkenburg, B., & Morrison, M. (Eds.). (2015).
Why more is different: Philosophical issues in condensed matter physics and complex systems. Springer.
Chibbaro, S., Rondoni, L., & Vulpiani, A. (2014).
Reductionism, emergence and levels of reality. Springer.
And Anderson's groundbreaking study on the autonomy of different levels of analysis and the non-derivability you assume:
More is Different
Although Anderson doesn't go far enough, you will find that the ability to derive higher levels from lower is a stronger condition than reductionism.
I've uploaded a scan of the paper from the volume it was published in.
Higher level structures can at best be in principle reduced to accord with the laws governing their constituent (lower level) parts. But they cannot be constructed from these laws. For example, in granular media the initial configuration state of a system that undergoes a transition, such as the formation of crystalline structure or that of a sandpile, does not violate any known laws of physics but the final structure cannot be "derived" from the initial. It is contingent upon the initial state and the collective (emergent) dynamics that are produced by forces acting on a system too interconnected to be understood solely in terms of its constituents.
The emergence of such collective structures that are irreducible and non-derivable from the dynamics of lower levels take on a qualitative difference for living systems, even for intracellular structures such as DNA. For living systems, functional emergence (which is non-physical) becomes necessary. Consider a model, simulation, or similar "realization" of a cell and the process of metabolic-repair, and let
f:
A→B be a function
"where
f is the process that takes input A and output B...The system Rosen uses for an example is the Metabolism-Repair or [M,R] system. The process, f, in this case stands for the entire metabolism goin on in an organism...The transition, f, which is being called metabolism, is a mapping taking some set of metabolites, A, into some set of products, B. What are the members of A? Really everything in the organism has to be included in A, and there has to be an implicit agreement that at least some of the members of A can enter the organism from its environment. What are the members of B? Many, if not all, of the memebers of A since the transitions in the reduced system are all strung together in the many intricate patterns or networks that make up the organism's metabolism. It also must be true that some members of B leave the organism as products of metabolism...In the context developed so far, the mapping, f, has a very special nature. It is a functional component of the system we are developing. A functional component has many interesting attributes. First of all,
it exists independent of the material parts that make it possible.
Reductionism has taught us that every thing in a real system can be expressed as a collection of material parts. This is not so in the case of functional components...Fragmentability is the aspect of systems that can be reduced to their material parts leaving recognizable material entities as the result. A system is not fragmentable is reducing it to its parts destroys something essential about that system. Since the crux of understanding a complex system had to do with identifying the context dependent functional components, they are by definition, not fragmentable". (emphasis added; italics in original)
Mikulecky, D. C. (2005). The Circle That Never Ends: Can Complexity be Made Simple? In D. Bonchev & D. H. Rouvray (Eds.).
Complexity in Chemistry, Biology, and Ecology (
Mathematical and Computational Chemistry). Springer.
"systems biology is concerned with the relationship between molecules and cells; it treats cells as organized, or organizing, molecular systems having both molecular and cellular properties. It is concerned with how life or the
functional properties thereof that are not yet in the molecules, emerge from the particular organization of and interactions between its molecular processes. It uses models to describe particular cells and generalizes over various cell types and organisms to arrive at new theories of cells as molecular systems. It is concerned with explaining and predicting cellular behaviour on the basis of molecular behaviour. It refers to function in ways that would not be permitted in physics. It addresses an essential minimum complexity exceeding that of any physical chemical system understood until now.
It shies away from reduction of the system under study to a collection of elementary particles. Indeed, it seems to violate many of the philosophical foundations of physics, often in ways unprecedented even by modern physics." (emphases added)
Boogerd, F., Bruggeman, F. J., Hofmeyr, J. H. S., & Westerhoff, H. V. (Eds.). (2007).
Systems biology: philosophical foundations. Elsevier.