The concepts of function, working principle and construction are also helpful for understanding biological evolution. For instance, Duncker (1994) has pointed out the importance of distinguishing between a functional and a causal level, which here connect to the function and working principle concepts, respectively. Functions are usually fulfilled by the interaction of several structural parts at the level of the construction, according to working principles. If von Uexküll (1928) had not previously introduced the term Wirkmal with a different meaning, it would be well suited to designate the structural embodiments of working principles (in German: Wirkprinzip). Alternatively, ‘constructional character’ and ‘constructional trait’ might be useful terms for designating those structures that play a role in fulfilling functions. This makes it possible to distinguish them from the character and trait concepts, which are not necessarily connected to a function, in phylogenetic homology research and from the descriptive term ‘morphem’ (Richter and Wirkner 2013, 2014).

Apart from overall functions (e.g. “to metabolise food”, “to sense the environment”), novel sub-functions can arise during evolution. For example, the predecessors of birds were at first unable to fly. However, once the forelimbs of these vertebrates had attained the characteristics enabling them to fly, the scope of the evolutionary variation of the ‘construction’ wing became limited (cf. Riedl 2000: 259). Not all constructions were now possible because of the implementation of a working principle, namely that by which wing profiles create a favourable ratio of lift to drag force within an air flow. The wings could thus not deviate far from a certain shape and the muscles, bones, etc. needed to be arranged in certain geometric relationships to each other. Hence, a bundle of constructional traits is strongly interconnected and cannot be changed to any great extent evolutionarily as long as these birds wish to fly. A functional burden (Riedl 1975, 1977) has emerged. Nevertheless, within the overall framework of a construction that is constrained by underlying working principles necessary to generate a certain function, a number of degrees of freedom exist with respect to the quantitative parameters (and their combinations) that are employed to realise the working principles and that, for example, generate certain mechanical lever conditions necessary for locomotion or feeding. Evolutionary morpho-physiological disparity can therefore be generated across the species within a clade without compromising function, a macroevolutionary principle well-known as the “many-to-one mapping of form to function” (Wainwright 2007, Wainwright et al. 2005).

Any (newly emerged) function channels the underlying morphological structures in a certain way by means of employed working principles. When a function is no longer relevant, its underlying structures may become released from their constraints and might either be reduced or evolve in the context of other functions. An example of such changes in function involves the cone-shaped pointed teeth of dolphins (personal communication, Wolfgang Maier, Tübingen). The phylogenetic ancestors of dolphins possessed, in their upper and lower jaws, teeth (tribosphenic molars) with cutting edges that fulfilled a specific function, namely “to cut up food”. Once this function was no longer needed, the working principle and the associated construction (cutting edges) could be abandoned. Extant dolphins can no longer cut up their food, with their teeth, and apparently no longer need this cutting function.

Functions can be preserved for a long time in phylogenesis, but they can be fulfilled by different working principles. During the pollination of plants, one function can be described as “to transport pollen grains to the target”. In the case of wind-pollinated flowers, this requires the pollen grains to be evenly distributed in the air and to hover until they reach a stigma with a good probability. Accordingly, the pollen grains must be ‘constructed’ following the (working) principles of aerodynamics. In pollination achieved by animal vectors, the same function is fulfilled differently and employs different working principles. The pollen grains no longer fly through the air by themselves, rendering the laws of aerodynamics meaningless for them. Instead, working principles now play a role that, for example, improve their attachment to the body of the animal pollinator. “To adhere to an animal’s body” thus becomes a new sub-function of “to transport pollen grains to the target”. Evolutionarily, both the transition from wind pollination to animal pollination and the reverse pathway are known (Wragg and Johnson 2011).

Functions of an organism can, as in the previous examples, relate directly to the external environment, which is the subject of research in the field of ecomorphology (cf. Reilly and Wainwright 1994, Betz 2008). However, functions and their implementations can also be largely independent of the external environment and can then be referred to as ‘internal functions’. Consider, for example, the function of a mammalian heart valve.

The consideration of function in the context of evolutionary research, the subject of analogy research (e.g. Koepcke 1974), is an under-represented field of research. This was recognised early on by Bertalanffy during his development of a systems view of biology: “It would be beneficial if, in the future, a greater experimental and theoretical interest were once again devoted to the problems of functional adaptation” (Bertalanffy 1932:320, own translation). This means that, in addition to homologies, analogies should also be investigated, because “similarities in form, particularly if they are not derived from the same evolutionary origins, appear to reveal certain rules with regard to the shaping of these forms [Bildungsgesetze]; these rules then dictate the course of certain evolutionary trends” (Bertalanffy 1933:75, own translation).

In order to link the notions of function, working principle and construction more comprehensively to evolutionary biology, we need to introduce the concept of optimisation as the next step.

The previously introduced concepts of function, working principle and construction can be connected via the design space to resolve optimisation problems. For this purpose, the field of engineering optimisation needs to be explored in some breadth.

Optimisation has been broadly defined as: “the efforts and processes of making a decision, a design, or a system as perfect, effective, or functional as possible” and more narrowly, as “the specific methodology, techniques, and procedures used to decide on the one specific solution in a defined set of possible alternatives that will best satisfy a selected criterion” (Licker et al. 2003). The decisions made with regard to the particular shape of Velcro® hooks and loops (Fig. 1) serve as an example. Even with given similar functions and working principles, constructions can vary to some extent. One reason for this might be optimisation processes with respect to different performance measures, whereby performance is defined as any quantitative measure of how well a function is accomplished (Irschick and Higham 2016). We therefore need to investigate the way in which the concepts of optimisation and optimum fit into the engineering design schema introduced above (Fig. 1).

In engineering, we can easily find out what has been optimised and with what objective(s), because these questions have guided the designer. In biological systems (e.g. the shapes of diatoms or the leaves of a tree), difficulties are encountered in determining whether and to what extent the systems have been optimised and the value of their adaptive peak. This requires solid knowledge of the function or biological role (Bock and Wahlert 1965) of the structures under consideration followed by comparative performance measurements (or theoretical modelling) in the biological context (e.g. Irschick and Higham 2016).

An instructive example for optimisation in engineering is the optimisation of a roof shape illustrated by Bletzinger et al. (1995). A rectangular ground plan is given and mathematically pre-defined. Bézier curves are used for the roof shape (Fig. 2). Two parameters (or design variables; Martins and Ning 2022) can be changed: the height of the roof in the middle of the short side of the ground plan (x₁) and the height of the roof in the centre of the rectangle (x₂). Otherwise, the roof is defined by the Bézier curves, which connect as higher or lower arches the longer sides of the ground plan. An engineer can, for instance, optimise the structure for minimum strain energy, which is equivalent to the minimum deformation of the roof. The roof would then look like a saddle (x₁ > x₂ in Fig. 2). Optimisation might also be aimed to achieve equal stress distribution, which would lead to a roof shape that is more similar to a dome (x₁ < x₂ as actually shown in Fig. 2). Alternatively, the roof could be optimised for minimum weight, resulting in a roof shape resembling part of a cylinder (x₁ = x₂ and both small in Fig. 2). From these considerations, we can clearly see that the designer has to choose the objective (and performance measure) to be achieved by the proposed optimisation. A subjective decision will thus be made between one of these three objectives.

The optimum and optimisation always refer to something particular: a particular objective (function). Moreover, those parameters that can be changed to achieve the objective have to be clear. For a simple case like a roof, this can easily be shown in a 3D diagram. The resulting design space in the roof example can have two axes for the two parameters that can be changed. The third (vertical) axis of the diagram indicates the optimisation objective by means of a quantitative measurement regarding the performance of the objective. In the simple two parameter case, one can consider a landscape (Fig. 3) in which one searches for the parameter combination with the highest peak, which represents the best performance, i.e. the optimum. The optimisation process can be considered as finding a quick way from a given (random) combination of parameters to the optimal one.

In general, optimisation is the determination of the parameter settings that correspond to a well-optimised system. This landscape depiction is useful when only two parameters are involved. In instances of more than two parameters, which is usually the case, the design space becomes a hyperspace. Here, we can usefully consider a space with clouds of greater or lesser density (high density representing high performance), with optimisation algorithms searching for the area of highest density.

From a biologist’s perspective, the question arises as to whether the engineering concepts of optimisation can be adapted to biological optimisation problems in real organisms. Cases of optimisation towards a single performance measure are difficult to find in biology. One example used by Nachtigall (1995) is the haematocrit, i.e. the volume percentage of red blood cells in the blood. This parameter has an optimum with respect to oxygen flow, i.e. the maximum amount of oxygen transported through the vessels per unit time. In cases of few erythrocytes, they move rapidly because of the resulting low viscosity of the blood but, together, they carry little oxygen. Many erythrocytes can carry a large amount of oxygen but they move slowly, because the viscosity of the blood is now high. The optimum of the haematocrit can be determined experimentally and theoretically and compared with haematocrit values found in organisms. These values have been shown to be at the optimum, i.e. maximum oxygen flow, in pigs and sheep (Nachtigall 1995).

Another important issue in engineering is multi-criteria (= multi-objective) optimisation (Alfaris 2010), which occurs once more than one criterion or objective needs to be achieved. Taking the roof example from above, we find that, if both equal stress distribution and minimum strain energy, are objectives, a good overall solution is hardly possible. Different methods are available to cope with such problems. For instance, the importance of the different objectives could be weighted with, say, 30% and 70% and expressed in only a single performance measure. Optimisation might then produce a mediocre performance for both criteria. Importantly, no single optimal solution can generally be achieved in multi-criteria optimisation. The choice as to which objective has to be weighted more highly involves the subjective decision of the designer and, furthermore, the engineer has to choose from a set of solutions that are only more or less optimal. In biology, multi-criteria optimisation problems are the rule. They do not lead to only one best ‘solution’ and usually involve suboptimal compromise solutions. For example, the shells of sea urchins might have been optimised with respect to multiple objectives involving not only mechanical stability, but also ontogenetic factors (Grun et al. 2016).

Since the engineering design schema (Fig. 1) turns out to be a useful conceptual framework, the question arises as to whether it can also be linked to optimisation problems in general (both engineering and biology). The conceptual levels that need to be considered here are function, working principle and construction, which can be easily linked to the design space (Fig. 4). A function such as “to attach an object reversibly” provides indication for an objective and a performance measure. A particular performance measure can be interpreted as a quantified specification of a function. For an engineering construction, one could, for instance, use a performance measurement such as “attachment force after 1000 cycles of attaching and detaching” for the function “to attach an object reversibly”. The performance of, for example, Velcro® tape will be considered satisfactory if it retains its high attachment force, even after 1000 cycles of use. For the construction of the biological concept generator, a somewhat different performance measure is conceivable, because the burr has to attach to the fur of the animal only once or twice in a reversible manner.

On the level of the working principle (Fig. 1), the connection to the design space is straightforward. Working principles directly indicate the parameters that can and need to be changed in the process of optimisation. For instance, parameters that can be changed in the Velcro® example include the diameter of a single hook, the diameter of the loop, the aspect ratio of the hook and material parameters such as Young’s modulus or the overall hook density.

The levels of both the function and working principle provide the performance criterion and the parameter dimensions of the design space. The working principle determines those parameters that have to be considered and altered during the optimisation process, whereas the function indirectly determines a measure for the performance that needs to be optimised.

Finally, at the construction level, the parameters determined by the working principles are set to specific values represented by a particular point in the design space. In the final construction, the parameters are set in such a way that performance is high, i.e. the construction fulfils the function well. Hence, the concept of optimisation via the design space can be seen as connecting the three levels of the scheme, namely function, working principle and construction (Fig. 4).

In this context, we should distinguish between the concepts of effectiveness and efficiency. Effectiveness refers to a quality, whereas efficiency can be quantified and can therefore, in specific cases, serve as a performance measure (cf. Fig. 3). A certain function can be fulfilled by means of alternative working principles and, usually, by a combination of them. In engineering, functions are commonly considered as being neutral with respect to the various working principles that can be used. “To attach an object reversibly” might be achieved via the working principles of mechanical interaction mentioned above in the Velcro® case but might also be based on alternative working principles such as those involving wet adhesion or the attraction of magnetic forces. They all indicate cause and effect relationships that can be implemented to achieve the particular function. Other working principles such as optical ones are not effective for the mentioned function, because they cannot contribute to fulfilling it. Hence, the mechanical, adhesive or magnetic working principles are all effective (i.e. they produce a useful effect with regard to fulfilling the function), whereas the optical working principles are not. The effect resulting from the employment of a specific working principle determines the highest level that the overall performance can achieve. Hence, the performance of a certain construction with respect to its function depends first on the selected working principles and can then, second, be further modified by fine-tuning the combination of the parameters that are indicated by the working principles (e.g. the material parameters in the Velcro® example). Effectiveness defines the overall quality of the working principle to be selected in order to fulfil a certain function (can a selected working principle produce a necessary effect at all?), whereby a working principle defines a quantifiable performance range that can potentially be achieved under the actual boundary conditions. For instance, for the conversion of solar (radiant) energy to electrical energy, working principles can be used that involve the heating of water in order to run steam turbines. Alternatively, a working principle that involves the photoelectric effect could be employed. Both these working principles come with different limitations to their efficiency, which in these cases can serve as a performance measure obtained here by dividing energy output by energy input. The theoretical maximum efficiency (η_max) using a steam turbine is given by the Carnot cycle and is limited by the lower (T₀) and upper temperature (T₁) in the cycle: η_max = 1 – T₀/T₁ (Grote and Antonsson 2008: 268). Using the photoelectric effect, the Shockley-Queisser limit allows for a maximum efficiency η_max of about 35% (Dau et al. 2019: 35). Functions can be achieved in differently efficient ways depending on the type of working principles used and the setting of their parameters.

Constructions using other working principles require the consideration of other parameters. For instance, the dimensions in the design space for the parameters that need to be optimised are different in the case of the steam turbine compared with photovoltaic construction. However, if the function stays the same, then the performance measure (in this case energy efficiency) also stays the same. Note that efficiency is one type of performance measure, others can be visual acuity, jumping hight, etc.

The described concept of effectiveness can also be applied to biological evolution in which, in theory, one working principle should be selected that allows the fulfilment of a function with high performance. Once this working principle is in action, its underlying parameter values are under selective pressure to further improve (optimise) the overall performance of the system employing the given working principle. The selection of a certain working principle in evolution might be attributable to chance (whereby exaptation (Gould and Vrba 1982) can be understood as working principles that are in place within an organism but with a function yet to come), to selective pressure (according to slight differences in the resulting performances of the working principles) or to certain environmental or organismal conditions.

For our understanding of macroevolution, we will find it instructive to analyse to what extent performance differences between clades have resulted from the selection of differently effective working principles or from the adjustment of more efficient parameters within the underlying working principles. For example, in predatory Stenus rove beetles (Coleoptera, Staphylinidae), a shift has occurred from the ancestral way of prey-capture by means of direct seizure with the mandibles to a novel (derived) prey-capture apparatus that involves the combination of new working principles such as the catapult-like protrusion of the elongated labium and the firm adhesive attachment of the prey to sticky cushions at the tip of the labium (reviewed in Betz et al. 2018). The selection of these novel working principles has probably made it possible for rove beetles to catch elusive prey such as springtails in a highly efficient and rapid manner, despite many life forms of these beetles being physiologically limited with respect to their overall agility and reaction ability (Betz 1998). The newly evolved labial prey-capture apparatus has been further adjusted within the selected working principles improving its efficiency (i.e. its performance) with regard to prey-capture success. These adjustments have involved an increase in the parameter values of the labium length (increasing its range), in the size of the sticky cushions and in the number of adhesive trichomes on their surface (increasing their adhesive strength) (Betz 1996, Koerner et al. 2017). Explanations concerning the origin of evolutionary novelties in biological organisms (e.g. Wagner 2010) should thus involve a closer inspection of the working principles that generate novel functions.

Biological evolution can lead to burdens or constraints (Riedl 1975, 1977, Held 2009). Once the working principles have been selected, they cannot easily be changed for others and they thereby determine the dimensions of the design space. Thus, effectiveness and performance (or in specific cases efficiency) can be limited, leading to ‘solutions’ in biology that are not, from an engineering point of view, optimal. However, within these constraints or burdens, the performance can increase during evolution, although it will not increase as much as it would have if an alternative working principle or new combinations of working principles had been initially used.

In engineering, as previously mentioned, a set of parameters of a construction is optimised towards multiple criteria during multi-criteria optimisation. All dimensions for the parameters stay the same but have to be optimised with respect to other (conflicting or concurrent) criteria. In biological evolution, another case of optimisation might also be relevant, here referred to as entangled optimisation (Fig. 5). This case of optimisation cannot be depicted in two dimensions easily. In Fig. 5, we hypothetically consider x₁ and x₂ parameters that are employed for optimisation with respect to performance measure A, with some optimal peak at a particular setting of x₁ and x₂. Now, let us consider that parameter x₁ is also involved in some other construction that is used with some other working principle that involves parameters x₁ and x₃ relevant for optimisation with regard to performance measure B. The optimal peak for B requires a certain setting of x₁ and x₃. However, the peaks of performance measures A and B each require different values of x₁. In such a case, it is in principle impossible to attain a maximum for both performance measures. This is different from multi-criterion optimisation, which concerns the same (and not only partly shared) parameters that are optimised with regard to different criteria. For sound localisation in vertebrates, for instance, it is advantageous to have the ear drums set far apart from each other, as this makes the detection of the direction of a sound more precise. However, the parameter of the distance between the ear drums is also connected to the parameter of head width. Since head width is subject to many other restrictions, it cannot be optimised (from an engineering perspective) to achieve an optimal solution for both requirements. To avoid such entanglements, the term module in the sense of largely independent units serving particular functions can be useful with respect to optimisation in both engineering and biology, as can, the term functional unit (as a mechanical unit or evolutionarily stable configuration) identified via shared functional interactions (Schwenk 2001). Such modules need to be distinguished from modules or modularity used in a gene-related sense (Futuyma 2007:517). “By modularity we mean a genotype-phenotype map in which there are few pleiotropic effects among characters serving different functions, with pleiotropic effects falling mainly among characters that are part of a single functional complex” (Wagner and Altenberg 1996).

In biology and engineering, certain parameters can be subject to neutral variation, i.e. certain values do not affect the final performance of the system, as has been introduced above in the context of many-to-one mapping. One example involves the branching patterns of blood vessels at the aortic arch (Popieluszko et al. 2017). Some humans have different patterns with regard to the branching of these vessels, i.e. the vessels have different geometrical parameters but with no major influence on performance (or health). An example from engineering refers to the elasticity of water pipelines. Plastic and steel have different Young’s modulus values and yet both materials can be used and perform well in pipes for transporting water under pressure.

To conclude, optimisation is a problem that is relevant for both engineering and biology. The engineer has to choose one or more objective(s) in accordance with a desired function and even has to consider multi-criteria optimisation problems that involve trade-off solutions. Hence, the envisaged optimum and the optimisation process are subject to the technical aims of the designer. For a biologist, the determination of the particular ‘objectives’ in the evolution of a particular structure can be challenging. Continuous adaptation to the environment is often important and might be indicated by the performance measures that can be investigated by biologists. For instance, the hook shapes on burs (geometric parameters) are well adapted enabling the wide dispersal of seeds. This links performance to fitness. Arnold (1983, 2003) has developed a concept that theoretically and practically connects particular morphological structures via their performance capacity to the fitness of a whole individual, i.e. an organism. Although the fitness consequences of subtle morphological differences between organisms are difficult to measure, we can assume that the measurable individual performance differences that result from morphological differences do indeed contribute to fitness.

So far, we have dealt with optimisation and the design space in a general way. Design spaces and their ‘landscapes’ can, however, be diverse, extending from very narrow parameter areas with high performance to broader areas with neutral variation. These areas of neutral variation are related to the concept of robustness, which will be explored next.

For this simplified engineering dovetail slide example (Fig. 7 right), imagine that part A is given without an additional adjustment device and part B has to be designed and manufactured anew. High performance would mean minimum backlash, although sliding is still possible. In this case, the two parameters for part B (x_1B, x_2B) must not vary to any large extent. This can be visualised as a single small peak in the design space (Fig. 6A, when x_1B and x_2B are depicted as x₁ and x₂). When the two parameters are well chosen, the resulting sliding performance can reach a high optimum. However, any minor perturbation, expressed as a change in either of the two parameters, will impair the performance and, thus, this kind of construction is not robust with respect to changes in x_1B and x_2B.

Now imagine a case in which x_1A and x_1B can be optimised together in the dovetail slides. When these parameters are changed simultaneously, several optimal solutions are possible. In the design space, this would resemble a ridge (Fig. 6B, when x_1A and x_1B are depicted as x₁ and x₂). The range of parameter value combinations for a good result is small but is larger than for the single peak case. A slight change in only one parameter would, however, lead to a lowered performance. Moreover, note that the ridge in the example of Fig. 6B is not of a constant height, which means that tiny (but still fitting) x_1A and x_1B lengths would easily lead to damage in part B, because a narrow dovetail would be prone to deformation and fracture. The ridge-like surface in the design space is typical for any engineering fit in which one part has to fit into another with small tolerance ranges (Grote and Antonsson 2008). Although ridge-like design spaces are common in technical devices (e.g. shafts in bearings), their robustness with respect to changes in the relevant parameters is only low.

Similar ‘constructions’ to the dovetail slide can be found in the mouthparts of various insects, for example, in the sucking mouthparts of true bugs (e.g. Wenk et al. 2010). The connections between the maxillae (Fig. 8) can be compared with the engineered dovetail slides, as they allow linear gliding past each other. However, the groove-and-tongue structures of the maxillae differ in detail from the engineered dovetail slides with respect to shape. One can compare the two constructions, i.e. the hemipteran and the engineered system and consider their robustness with respect to changes in their geometric parameters. In the hemipteran stylets, the curvatures of the groove-and-tongue structures probably allow for a wider range of variation in geometric parameters while still giving good performance with regard to sliding in a precise manner. Compared with the engineering construction, no flat surfaces glide past each other. Hence, more than two parameters can vary. Potentially, material compliance can also allow for adjustments of length measures. Unlike technical constructions, biological materials are often soft and elastic, so that the relevant material parameters such as Young’s modulus might allow for robust constructions that still work well, even when, for example, a radius in the groove-and-tongue structures changes.

A broad optimum would, of course, be much better than a small peak or a narrow ridge in the design space. To illustrate this, first consider a piston pump in engineering (Fig. 9A) in which a piston moves back and forth within a cylinder, thereby changing the volume within the cylinder periodically. An inlet and an outlet valve open in such a way that the liquid moves in one direction only. In engineering, both pistons and cylinders are often made of metal, and the diameters of both have to be chosen in such a way that movement is possible, despite the gap between them being small in order to prevent liquid escaping between the piston and cylinder. Such an arrangement is not very robust with respect to changes of the parameters “outer diameter of the piston” (considered as x₁) and “inner diameter of the cylinder” (considered as x₂) (resembling the design space in Fig. 6B) or to deformations that result from impacts from the exterior. Salivary pumps in hemipterans have an arrangement different from that of technical piston pumps (Fig. 9B). The materials are not rigid and their functional elements are integrated and not separate. The gap between the ‘piston’ (P in Fig. 9B) and the ‘cylinder’ (Cu in Fig. 9B) is wide and the cylinder does not need to be sealed. Hence, even if the diameter of the piston is slightly changed, it can still move, without the performance of pumping with high efficiency being affected, since the piston does not touch the cylinder as in the engineering case. Furthermore, the engineering parts have to have constant diameters throughout. This is not necessary in the salivary pump, because the ‘cylinder’ is not used as guide for the ‘piston’. Hence, no direct equivalent to x₁ and x₂ of the engineering case can be found for the salivary pump. Moreover, changes in the inner diameter of the cylinder of the bug would not result in a significant drop in performance. Hence, a flat optimum would result if both these parameters were plotted (from a particular cross section as indicated by the red dotted line in Fig. 9B) in the design space (Fig. 6C). Such an optimum is much more robust with respect to changes, in this case of diameters, than is the case for the single optimum peak. However, with a rigid metal construction, one can produce higher pressures that are not achievable by the biological salivary pump ‘construction’, because material deformations attributable to changes in pressure are low. In engineering, diaphragm pumps (Eifler et al. 2009) use soft parts similar to those of the hemipteran salivary pump. Conversely, to our knowledge, nothing similar to the conventional engineering piston pump exists in biological ‘constructions’.

The vertebrate heart (Fig. 9C) also uses periodical volume contractions for pumping, but in a different way. No piston is involved, but rather the whole muscular surrounding of the blood volume contracts and the movements of the valves lead to a flow of blood in one direction only. From an engineering perspective, the many (morphometric) length measures (e.g. diameters of the left ventricle in various cross sections) also do not seem to be highly constrained, i.e. limited to small ranges. Whereas the piston would no longer move in case of a small diameter reduction of, say, 0.1 mm of the cylinder in an engineered piston pump, a similar reduction in the size of a ventricle might not greatly alter the overall performance of the heart. Therefore, in the design space for many size parameters of the heart, a flat optimum (Fig. 6C) can be expected, similar to that of the hemipteran salivary pump. A supporting fact for this claim is that, although hearts shrink during ageing, this does not affect their performance, as investigated for the size of the left ventricle (Peverill 2021).

A comparison of technical scissors and cutting edges in biological teeth serves as another example. The two cutting edges of scissors are rigidly connected via a screw that does not allow other than one degree of freedom to work properly. If any of the geometric parameters changes (e.g. caused by a loose connection or the deformation of the blades), the cutting performance deteriorates. Analogous to technical scissors, many vertebrates have teeth with edges, whereby the edges of two opposing teeth are used to cut up food (Maier 2017, 2020). In contrast to the scissors, the cutting edges are not necessarily arranged with high precision. For example, precise occlusion is lacking in many herbivorous lizards that cut vegetation with laterally multicuspate teeth. The position and the form of the upper and lower teeth in this example can vary widely before the cropping performance declines (Schwenk 2001). Furthermore, although teeth are hard and rigid, they are connected to the jaw bones via an elastic periodontium (Lippert 2003:602), which makes slight movements of the teeth relative to the jaw possible. Moreover, the lower jaw can be moved in various directions as a whole, directing the cutting edges to those places in which they are needed. Hence, this example shows that some morphological parameters can vary widely without affecting the performance relevant for food ingestion. However, having more than one degree of freedom of the scissors, the movements of jaws in different directions have to be controlled in a more sophisticated manner via various muscles. Additionally, much more precise and straight cuts can be made with scissors, but such precision is hardly needed by organisms.

In general, a high flat elevation in a design space with two parameters (x_1, x₂) indicates an optimal area that is robust (Fig. 6C), i.e. with a high plateau, so that parameters can be changed within wider limits without compromising performance. The size of the plateau area can be used as a measure of robustness, because it is quantifiable. In the presented design space diagrams (Fig. 6), a search for robust solutions entails one looking for flat maximum areas in the design space. In multidimensional design spaces, the landscape analogy no longer holds although the basic concept remains the same. In the mentioned engineering examples (dovetail slide, technical piston pump), such a flat optimum area is non-existent, because of the way that they are constructed. On the other hand, the biological example of the salivary pump allows a flat optimum for considerations of, for example, the diameters of piston and cylinder.

Examples for robustness are not limited to morphological features but are probably widespread in biology. For example, robustness plays an important role in the field of systems biology in which dynamic behaviours are investigated, mainly on the molecular scale (e.g. Alon 2007). For example, particular arrangements of signal transduction pathways on the molecular level can contribute to robustness against noise (Paul et al. 2016). Biochemical reactions arranged in a sequence need especially fine-tuned values of parameters (reaction rate constants) to work well in transmitting a signal, for example, from the outside of a cell to its nucleus. Theoretical considerations show that such arrangements are not robust with respect to slight changes in parameter values. However, in real biological systems, we find that network-like arrangements of chemical reactions filter out noise without the need for fine-tuned parameters in order to perform well (Paul et al. 2016). The possibility for a certain degree of “sloppiness” in the parameter values without negative consequences for the overall performance seems to be an important feature in real biochemical systems and their models (Gutenkunst et al. 2007). Chain-like versus net-like arrangements of chemical reactions, which can be seen as realising different working principles, have different “connectivities” or topological configurations. When the performance is ‘optimised’ (by evolution or in computer simulations), the reaction rate constants can be considered as parameters of the design space. Other arrangements of the chemical reactions lead to different design spaces in which, in the robust case, the optimum is large, since (sloppy) parameters can vary widely but still maintain a good performance.

In an even more general perspective, going beyond morphological structures and systems biology, the approach developed here may turn out to be useful for the broader debate on robustness. Up to now, no commonly accepted definition exists for robustness. The illustration via the design space (Fig. 6) might, however, be useful for its more precise conceptualisation in general. Robustness has been characterised as the “ability to withstand attacks, perturbations and offences without being disrupted or heavily modified” (Bertolaso et al. 2018:2). According to a more specific definition from systems biology (and engineering), robustness is a “property that allows a system to maintain its functions despite external and internal perturbations” (Kitano 2004). Since robustness represents a fundamental system level phenomenon, it is also relevant for the central organism concept in biology (Bertolaso et al. 2018:2). Organisms need to be robust with respect to changes or disturbances from both the exterior (e.g. temperature changes) and the interior (e.g. physiological changes), i.e. they still need to be able to perform their functions well. The above definitions comprise both external and internal perturbations, which can be visualised by changes of the parameter values in the design space, together with possible alterations of the resulting performance. In a macroevolutionary context, perturbations can be interpreted with respect to long-term changes of organismal ‘constructions’. A design space with robust areas (flat optimum) would thereby indicate the scope of variation possible without compromising performance (e.g. swimming performance of cephalopods with respect to shell parameters, as analysed by McGhee 2007).

The connection between robustness and evolvability plays an important role in evolutionary biology (Wagner 2005). Evolvability can roughly be considered as the ability or capacity to evolve or, more specifically, the capacity to adapt to changing conditions (Futuyma 2007:517). A paradox appears that, in evolution, organisms need to be robust but evolvable at the same time. This raises several problems that have been addressed in recent years (Wagner 2005, 2016). On the level of the phenotype, a design space with connected (large) areas of high performance facilitates the ability to evolve, because paths through valleys of low performance can be avoided during evolution. The combination of working principles ensuring that such paths with high performance throughout the design space are possible is therefore beneficial or even a pre-condition for adaptive evolutionary change. The previously mentioned principle of many-to-one mapping of form to function also implies that, in the design space, several parameter combinations of the underlying working principles result in equally high-performance values (Wainwright 2007, Wainwright et al. 2005). This can facilitate the evolvability of forms on the condition that these high-performance areas are not separated by deep valleys in the design space. The combination of parameters (determined by the working principles) of the design space determines the character of the ‘landscape’ and, hence, constrains the possible evolutionary pathways. The design space can be explored more widely when connecting corridors (Fig. 6D) between (flat) optima are available, i.e. when pathways of high performance are present. Such corridors with relatively high-performance values are more important in biological evolution than in engineering, because engineers do not have to change their constructions from one product generation to the next in a step-by-step manner, but can design them anew. Unlike in engineering, every biological organism during the process of phylogenesis has to be viable and able to reproduce. In this case, the realised constructions and the used working principles have to allow for gradual changes from one robust optimum to another.

Wagner (2016) has shown that, in high dimensional hyperspaces, many feasible, i.e. viable, connections are possible for the phenotypes that he has investigated on a molecular level, although his approach can probably also be extended to the morphological level, as considered here. On the molecular phenotypic level, one of Wagner’s questions is: how many proteins (i.e. amino acid sequences) can perform one particular function (Wagner 2016:117)? Astonishing high numbers of such proteins have been detected that, surprisingly, are connected by a network of uninterrupted pathways in which one amino acid is changed at a time (Wagner 2016:123). Such networks in high dimensional spaces cannot be visualised in 2D or 3D diagrams but they can be described with mathematical tools (Reidys et al. 1997). In the mathematically defined space that represents huge numbers of different proteins, the steps consisting of one amino acid change have been found to form pathways and networks in which the functions and performances of the altered proteins are preserved. In plant and animal morphology, analogous features may be found in the design space with quantitative parameters for morphological structures. More than one corridor of high performance, as labelled in Fig. 6D, might be available when considering the many parameters that are often necessary to describe morphological structures that need to be represented by a multidimensional design space. The many corridors could then be connected and might form a network analogous to that found for proteins. This would significantly contribute to the evolvability of morphological structures, because many regions in the design space could be reached without drops in performance. Similar to the more mechanical examples discussed above, the shape of a protein (e.g. an enzyme; in this case, the molecular shape results from 2D and 3D folding) is a decisive factor for its being suited to fulfil a specific function. We can thus assume that analogous hyperspace characteristics, including many corridors, also exist on higher morphological levels, if working principles are utilized in a suitable manner.

Importantly, one central difference exists between engineering optimisation and adaptiogenesis in biological evolution. Whereas the dimensions of the design space are given and do not change in conventional engineering optimisation problems, new dimensions can emerge during phylogenesis. If, for example, morphological novelties evolve, the dimensions of the design space increase. Alternatively, if organisms reduce certain structures (for example, during vestigialization), these dimensions decrease, which means that no fixed, unalterable and pre-defined design space exists in which organisms evolve (Longo 2018).