Genotype networks, innovation, and robustness in sulfur metabolism

Quantum aspects of evolution: a contribution towards evolutionary explorations of genotype networks via quantum walks

Quantum biology seeks to explain biological phenomena via quantum mechanisms, such as enzyme reaction rates via tunnelling and photosynthesis energy efficiency via coherent superposition of states. However, less effort has been devoted to study the role of quantum mechanisms in biological evolution. In this paper, we used transcription factor networks with two and four different phenotypes, and used classical random walks (CRW) and quantum walks (QW) to compare network search behaviour and efficiency at finding novel phenotypes between CRW and QW. In the network with two phenotypes, at temporal scales comparable to decoherence time TD, QW are as efficient as CRW at finding new phenotypes. In the case of the network with four phenotypes, the QW had a higher probability of mutating to a novel phenotype than the CRW, regardless of the number of mutational steps (i.e. 1, 2 or 3) away from the new phenotype. Before quantum decoherence, the QW probabilities become higher turning the QW effectively more efficient than CRW at finding novel phenotypes under different starting conditions. Thus, our results warrant further exploration of the QW under more realistic network scenarios (i.e. larger genotype networks) in both closed and open systems (e.g. by considering Lindblad terms).

Adding levels of complexity enhances robustness and evolvability in a multilevel genotype-phenotype map

Robustness and evolvability are the main properties that account for the stability and accessibility of phenotypes. They have been studied in a number of computational genotype-phenotype maps. In this paper, we study a metabolic genotype-phenotype map defined in toyLIFE, a multilevel computational model that represents a simplified cellular biology. toyLIFE includes several levels of phenotypic expression, from proteins to regulatory networks to metabolism. Our results show that toyLIFE shares many similarities with other seemingly unrelated computational genotype-phenotype maps. Thus, toyLIFE shows a high degeneracy in the mapping from genotypes to phenotypes, as well as a highly skewed distribution of phenotypic abundances. The neutral networks associated with abundant phenotypes are highly navigable, and common phenotypes are close to each other in genotype space. All of these properties are remarkable, as toyLIFE is built on a version of the HP protein-folding model that is neither robust nor evolvable: phenotypes cannot be mutually accessed through point mutations. In addition, both robustness and evolvability increase with the number of genes in a genotype. Therefore, our results suggest that adding levels of complexity to the mapping of genotypes to phenotypes and increasing genome size enhances both these properties.

On the networked architecture of genotype spaces and its critical effects on molecular evolution

Evolutionary dynamics is often viewed as a subtle process of change accumulation that causes a divergence among organisms and their genomes. However, this interpretation is an inheritance of a gradualistic view that has been challenged at the macroevolutionary, ecological and molecular level. Actually, when the complex architecture of genotype spaces is taken into account, the evolutionary dynamics of molecular populations becomes intrinsically non-uniform, sharing deep qualitative and quantitative similarities with slowly driven physical systems: nonlinear responses analogous to critical transitions, sudden state changes or hysteresis, among others. Furthermore, the phenotypic plasticity inherent to genotypes transforms classical fitness landscapes into multiscapes where adaptation in response to an environmental change may be very fast. The quantitative nature of adaptive molecular processes is deeply dependent on a network-of-networks multilayered structure of the map from genotype to function that we begin to unveil.

Adaptive multiscapes: an up-to-date metaphor to visualize molecular adaptation

Background

Wright’s metaphor of the fitness landscape has shaped and conditioned our view of the adaptation of populations for almost a century. Since its inception, and including criticism raised by Wright himself, the concept has been surrounded by controversy. Among others, the debate stems from the intrinsic difficulty to capture important features of the space of genotypes, such as its high dimensionality or the existence of abundant ridges, in a visually appealing two-dimensional picture. Two additional currently widespread observations come to further constrain the applicability of the original metaphor: the very skewed distribution of phenotype sizes (which may actively prevent, due to entropic effects, the achievement of fitness maxima), and functional promiscuity (i.e. the existence of secondary functions which entail partial adaptation to environments never encountered before by the population).

Results

Here we revise some of the shortcomings of the fitness landscape metaphor and propose a new “scape” formed by interconnected layers, each layer containing the phenotypes viable in a given environment. Different phenotypes within a layer are accessible through mutations with selective value, while neutral mutations cause displacements of populations within a phenotype. A different environment is represented as a separated layer, where phenotypes may have new fitness values, other phenotypes may be viable, and the same genotype may yield a different phenotype, representing genotypic promiscuity. This scenario explicitly includes the many-to-many structure of the genotype-to-phenotype map. A number of empirical observations regarding the adaptation of populations in the light of adaptive multiscapes are reviewed.

Conclusions

Several shortcomings of Wright’s visualization of fitness landscapes can be overcome through adaptive multiscapes. Relevant aspects of population adaptation, such as neutral drift, functional promiscuity or environment-dependent fitness, as well as entropic trapping and the concomitant impossibility to reach fitness peaks are visualized at once. Adaptive multiscapes should aid in the qualitative understanding of the multiple pathways involved in evolutionary dynamics.

Reviewers

This article was reviewed by Eugene Koonin and Ricard Solé.

The potential for non-adaptive origins of evolutionary innovations in central carbon metabolism

BACKGROUND

Biological systems are rife with examples of pre-adaptations or exaptations. They range from the molecular scale - lens crystallins, which originated from metabolic enzymes - to the macroscopic scale, such as feathers used in flying, which originally served thermal insulation or waterproofing. An important class of exaptations are novel and useful traits with non-adaptive origins. Whether such origins could be frequent cannot be answered with individual examples, because it is a question about a biological system's potential for exaptation. We here take a step towards answering this question by analyzing central carbon metabolism, and novel traits that allow an organism to survive on novel sources of carbon and energy. We have previously applied flux balance analysis to this system and predicted the viability of 10¹⁵ metabolic genotypes on each of ten different carbon sources.

RESULTS

We here use this exhaustive genotype-phenotype map to ask whether a central carbon metabolism that is viable on a given, focal carbon source C - the equivalent of an adaptation in our framework - is usually or rarely viable on one or more other carbon sources C _new - a potential exaptation. We show that most metabolic genotypes harbor potential exaptations, that is, they are viable on one or more carbon sources C _new . The nature and number of these carbon sources depends on the focal carbon source C itself, and on the biochemical similarity between C and C _new . Moreover, metabolisms that show a higher biomass yield on C, and that are more complex, i.e., they harbor more metabolic reactions, are viable on a greater number of carbon sources C _new .

CONCLUSIONS

A high potential for exaptation results from correlations between the phenotypes of different genotypes, and such correlations are frequent in central carbon metabolism. If they are similarly abundant in other metabolic or biological systems, innovations may frequently have non-adaptive ("exaptive") origins.

Phenotypic innovation through recombination in genome-scale metabolic networks

Recombination is an important source of metabolic innovation, especially in prokaryotes, which have evolved the ability to survive on many different sources of chemical elements and energy. Metabolic systems have a well-understood genotype-phenotype relationship, which permits a quantitative and biochemically principled understanding of how recombination creates novel phenotypes. Here, we investigate the power of recombination to create genome-scale metabolic reaction networks that enable an organism to survive in new chemical environments. To this end, we use flux balance analysis, an experimentally validated computational method that can predict metabolic phenotypes from metabolic genotypes. We show that recombination is much more likely to create novel metabolic abilities than random changes in chemical reactions of a metabolic network. We also find that phenotypic innovation is more likely when recombination occurs between parents that are genetically closely related, phenotypically highly diverse, and viable on few rather than many carbon sources. Survival on a new carbon source preferentially involves reactions that are superessential, that is, essential in many metabolic networks. We validate our observations with data from 61 reconstructed prokaryotic metabolic networks. Our systematic and quantitative analysis of metabolic systems helps understand how recombination creates innovation.

Structuring evolution: biochemical networks and metabolic diversification in birds

Background

Recurrence and predictability of evolution are thought to reflect the correspondence between genomic and phenotypic dimensions of organisms, and the connectivity in deterministic networks within these dimensions. Direct examination of the correspondence between opportunities for diversification imbedded in such networks and realized diversity is illuminating, but is empirically challenging because both the deterministic networks and phenotypic diversity are modified in the course of evolution. Here we overcome this problem by directly comparing the structure of a “global” carotenoid network – comprising of all known enzymatic reactions among naturally occurring carotenoids – with the patterns of evolutionary diversification in carotenoid-producing metabolic networks utilized by birds.

Results

We found that phenotypic diversification in carotenoid networks across 250 species was closely associated with enzymatic connectivity of the underlying biochemical network – compounds with greater connectivity occurred the most frequently across species and were the hotspots of metabolic pathway diversification. In contrast, we found no evidence for diversification along the metabolic pathways, corroborating findings that the utilization of the global carotenoid network was not strongly influenced by history in avian evolution.

Conclusions

The finding that the diversification in species-specific carotenoid networks is qualitatively predictable from the connectivity of the underlying enzymatic network points to significant structural determinism in phenotypic evolution.

Electronic supplementary material

The online version of this article (doi:10.1186/s12862-016-0731-z) contains supplementary material, which is available to authorized users.

Evolutionary escape on complex genotype-phenotype networks

We study the problem of evolutionary escape that is the process whereby a population under sudden changes in the selective pressures acting upon it try to evade extinction by evolving from previously well-adapted phenotypes to those that are favoured by the new selective pressure. We perform a comparative analysis between results obtained by modelling genotype space as a regular hypercube (H-graphs), which is the scenario considered in previous work on the subject, to those corresponding to a complex genotype-phenotype network (B-graphs). In order to analyse the properties of the escape process on both these graphs, we apply a general theory based on multi-type branching processes to compute the evolutionary dynamics and probability of escape. We show that the distribution of distances between phenotypes in B-graphs exhibits a much larger degree of heterogeneity than in H-graphs. This property, one of the main structural differences between both types of graphs, causes heterogeneous behaviour in all results associated to the escape problem. We further show that, due to the heterogeneity characterising escape on B-graphs, escape probability can be underestimated by assuming a regular hypercube genotype network, even if we compare phenotypes at the same distance in H-graphs. Similarly, it appears that the complex structure of B-graphs slows down the rate of escape.

Tradeoff between robustness and elaboration in carotenoid networks produces cycles of avian color diversification

Background

Resolution of the link between micro- and macroevolution calls for comparing both processes on the same deterministic landscape, such as genomic, metabolic or fitness networks. We apply this perspective to the evolution of carotenoid pigmentation that produces spectacular diversity in avian colors and show that basic structural properties of the underlying carotenoid metabolic network are reflected in global patterns of elaboration and diversification in color displays. Birds color themselves by consuming and metabolizing several dietary carotenoids from the environment. Such fundamental dependency on the most upstream external compounds should intrinsically constrain sustained evolutionary elongation of multi-step metabolic pathways needed for color elaboration unless the metabolic network gains robustness – the ability to synthesize the same carotenoid from an additional dietary starting point.

Results

We found that gains and losses of metabolic robustness were associated with evolutionary cycles of elaboration and stasis in expressed carotenoids in birds. Lack of metabolic robustness constrained lineage’s metabolic explorations to the immediate biochemical vicinity of their ecologically distinct dietary carotenoids, whereas gains of robustness repeatedly resulted in sustained elongation of metabolic pathways on evolutionary time scales and corresponding color elaboration.

Conclusions

The structural link between length and robustness in metabolic pathways may explain periodic convergence of phylogenetically distant and ecologically distinct species in expressed carotenoid pigmentation; account for stasis in carotenoid colors in some ecological lineages; and show how the connectivity of the underlying metabolic network provides a mechanistic link between microevolutionary elaboration and macroevolutionary diversification.

Reviewers

This article was reviewed by Junhyong Kim, Eugene Koonin, and Fyodor Kondrashov. For complete reports, see the Reviewers’ reports section.

Electronic supplementary material

The online version of this article (doi:10.1186/s13062-015-0073-6) contains supplementary material, which is available to authorized users.

Exhaustive Analysis of a Genotype Space Comprising 10(15 )Central Carbon Metabolisms Reveals an Organization Conducive to Metabolic Innovation

All biological evolution takes place in a space of possible genotypes and their phenotypes. The structure of this space defines the evolutionary potential and limitations of an evolving system. Metabolism is one of the most ancient and fundamental evolving systems, sustaining life by extracting energy from extracellular nutrients. Here we study metabolism’s potential for innovation by analyzing an exhaustive genotype-phenotype map for a space of 10¹⁵ metabolisms that encodes all possible subsets of 51 reactions in central carbon metabolism. Using flux balance analysis, we predict the viability of these metabolisms on 10 different carbon sources which give rise to 1024 potential metabolic phenotypes. Although viable metabolisms with any one phenotype comprise a tiny fraction of genotype space, their absolute numbers exceed 10⁹ for some phenotypes. Metabolisms with any one phenotype typically form a single network of genotypes that extends far or all the way through metabolic genotype space, where any two genotypes can be reached from each other through a series of single reaction changes. The minimal distance of genotype networks associated with different phenotypes is small, such that one can reach metabolisms with novel phenotypes – viable on new carbon sources – through one or few genotypic changes. Exceptions to these principles exist for those metabolisms whose complexity (number of reactions) is close to the minimum needed for viability. Increasing metabolic complexity enhances the potential for both evolutionary conservation and evolutionary innovation.

Genotype-phenotype mapping is one of the ultimate goals of computational systems biology, and can provide new insights into the function and evolution of biological systems. We present a comprehensive genotype-phenotype map for a space of metabolic genotypes that comprises more than 10¹⁵ central carbon metabolisms. Only one in a million of these metabolisms can sustain life on any one of 10 carbon sources we consider, but these viable metabolisms form connected genotype networks that extend far through genotype space. In addition, they render multiple novel metabolic phenotypes in their immediate neighborhood accessible through small evolutionary changes that require only the alteration of single metabolic reactions. The map we construct reveals an organization of core metabolism that simultaneously facilitates evolutionary conservation of existing metabolic phenotypes, and the origination of novel metabolic traits that allow viability on novel carbon sources. Such metabolic innovation is essential, particularly for organisms that experience unexpected environmental changes, and that explore or invade new habitats.

Surviving evolutionary escape on complex genotype-phenotype networks

We study the problem of evolutionary escape and survival of cell populations with a genotype-phenotype structure. We refer to evolutionary escape as the process where a cell of a given ill-adapted population to reach a well-adapted phenotype. Similarly, survival refers to the dynamics of the population once the escape phenotype has been reached. The aim of this paper is to analyse the influence of topological properties associated to robustness and evolvability on the probability of escape and on the probability of survival. In order to explore these issues, we formulate a population dynamics model, consisting of a multi-type time-continuous branching process, where types are associated to genotypes and their birth and death probabilities depend on the associated phenotype (non-escape or escape). We exploit the separation of time scales introduced by the the difference in reproductive ratios between the ill-adapted phenotypes and the escape phenotype. Two dynamical regimes emerge: a fast-decaying regime associated to the escape process itself, and a slow regime which corresponds to the survival dynamics of the population once the escape phenotype has been reached. We exploit this separation of time scales to analyse the topological factors which determine escape and survival probabilities. We show that, while the escape probability depends on the degree of escape phenotype, the probability of survival is essentially determined by its robustness, measured in terms of a weighted clustering coefficient.

The topology of robustness and evolvability in evolutionary systems with genotype-phenotype map

In this paper we formulate a topological definition of the concepts of robustness and evolvability. We start our investigation by formulating a multiscale model of the evolutionary dynamics of a population of cells. Our cells are characterised by a genotype-phenotype map: their chances of survival under selective pressure are determined by their phenotypes, whereas the latter are determined their genotypes. According to our multiscale dynamics, the population dynamics generates the evolution of a genotype-phenotype network. Our representation of the genotype-phenotype network is similar to previously described ones, but has a novel element, namely, our network contains two types of nodes: genotype and phenotype nodes. This network representation allows us to characterise robustness and evolvability in terms of its topological properties: phenotypic robustness by means of the clustering coefficient of the phenotype nodes, and evolvability as the emergence of giant connected component which allows navigation between phenotypes. This topological definition of evolvability allows us to characterise the so-called robustness of evolvability, which is defined in terms of the robustness against attack (i.e. edge removal) of the giant connected component. An investigation of the factors that affect the robustness of evolvability shows that phenotypic robustness and the cryptic genetic variation are key to the integrity of the ability to innovate. These results fit within the framework of a number of models which point out that robustness favours rather than hindering evolvability. We further show that the corresponding phenotype network, defined as the one-component projection of the whole genotype-phenotype network, exhibits the small-world phenomenon, which implies that in this type of evolutionary system the rate of adaptability is enhanced.

Historical contingency and the gradual evolution of metabolic properties in central carbon and genome-scale metabolisms

BACKGROUND

A metabolism can evolve through changes in its biochemical reactions that are caused by processes such as horizontal gene transfer and gene deletion. While such changes need to preserve an organism's viability in its environment, they can modify other important properties, such as a metabolism's maximal biomass synthesis rate and its robustness to genetic and environmental change. Whether such properties can be modulated in evolution depends on whether all or most viable metabolisms - those that can synthesize all essential biomass precursors - are connected in a space of all possible metabolisms. Connectedness means that any two viable metabolisms can be converted into one another through a sequence of single reaction changes that leave viability intact. If the set of viable metabolisms is disconnected and highly fragmented, then historical contingency becomes important and restricts the alteration of metabolic properties, as well as the number of novel metabolic phenotypes accessible in evolution.

RESULTS

We here computationally explore two vast spaces of possible metabolisms to ask whether viable metabolisms are connected. We find that for all but the simplest metabolisms, most viable metabolisms can be transformed into one another by single viability-preserving reaction changes. Where this is not the case, alternative essential metabolic pathways consisting of multiple reactions are responsible, but such pathways are not common.

CONCLUSIONS

Metabolism is thus highly evolvable, in the sense that its properties could be fine-tuned by successively altering individual reactions. Historical contingency does not strongly restrict the origin of novel metabolic phenotypes.

A latent capacity for evolutionary innovation through exaptation in metabolic systems

Some evolutionary innovations may originate non-adaptively as exaptations, or pre-adaptations, which are by-products of other adaptive traits. Examples include feathers, which originated before they were used in flight, and lens crystallins, which are light-refracting proteins that originated as enzymes. The question of how often adaptive traits have non-adaptive origins has profound implications for evolutionary biology, but is difficult to address systematically. Here we consider this issue in metabolism, one of the most ancient biological systems that is central to all life. We analyse a metabolic trait of great adaptive importance: the ability of a metabolic reaction network to synthesize all biomass from a single source of carbon and energy. We use novel computational methods to sample randomly many metabolic networks that can sustain life on any given carbon source but contain an otherwise random set of known biochemical reactions. We show that when we require such networks to be viable on one particular carbon source, they are typically also viable on multiple other carbon sources that were not targets of selection. For example, viability on glucose may entail viability on up to 44 other sole carbon sources. Any one adaptation in these metabolic systems typically entails multiple potential exaptations. Metabolic systems thus contain a latent potential for evolutionary innovations with non-adaptive origins. Our observations suggest that many more metabolic traits may have non-adaptive origins than is appreciated at present. They also challenge our ability to distinguish adaptive from non-adaptive traits.

Superessential reactions in metabolic networks

The metabolic genotype of an organism can change through loss and acquisition of enzyme-coding genes, while preserving its ability to survive and synthesize biomass in specific environments. This evolutionary plasticity allows pathogens to evolve resistance to antimetabolic drugs by acquiring new metabolic pathways that bypass an enzyme blocked by a drug. We here study quantitatively the extent to which individual metabolic reactions and enzymes can be bypassed. To this end, we use a recently developed computational approach to create large metabolic network ensembles that can synthesize all biomass components in a given environment but contain an otherwise random set of known biochemical reactions. Using this approach, we identify a small connected core of 124 reactions that are absolutely superessential (that is, required in all metabolic networks). Many of these reactions have been experimentally confirmed as essential in different organisms. We also report a superessentiality index for thousands of reactions. This index indicates how easily a reaction can be bypassed. We find that it correlates with the number of sequenced genomes that encode an enzyme for the reaction. Superessentiality can help choose an enzyme as a potential drug target, especially because the index is not highly sensitive to the chemical environment that a pathogen requires. Our work also shows how analyses of large network ensembles can help understand the evolution of complex and robust metabolic networks.

The influence of assortativity on the robustness of signal-integration logic in gene regulatory networks

Gene regulatory networks (GRNs) drive the cellular processes that sustain life. To do so reliably, GRNs must be robust to perturbations, such as gene deletion and the addition or removal of regulatory interactions. GRNs must also be robust to genetic changes in regulatory regions that define the logic of signal-integration, as these changes can affect how specific combinations of regulatory signals are mapped to particular gene expression states. Previous theoretical analyses have demonstrated that the robustness of a GRN is influenced by its underlying topological properties, such as degree distribution and modularity. Another important topological property is assortativity, which measures the propensity with which nodes of similar connectivity are connected to one another. How assortativity influences the robustness of the signal-integration logic of GRNs remains an open question. Here, we use computational models of GRNs to investigate this relationship. We separately consider each of the three dynamical regimes of this model for a variety of degree distributions. We find that in the chaotic regime, robustness exhibits a pronounced increase as assortativity becomes more positive, while in the critical and ordered regimes, robustness is generally less sensitive to changes in assortativity. We attribute the increased robustness to a decrease in the duration of the gene expression pattern, which is caused by a reduction in the average size of a GRN's in-components. This study provides the first direct evidence that assortativity influences the robustness of the signal-integration logic of computational models of GRNs, illuminates a mechanistic explanation for this influence, and furthers our understanding of the relationship between topology and robustness in complex biological systems.

The role of robustness in phenotypic adaptation and innovation

Phenotypes that vary in response to DNA mutations are essential for evolutionary adaptation and innovation. Therefore, it seems that robustness, a lack of phenotypic variability, must hinder adaptation. The main purpose of this review is to show why this is not necessarily correct. There are two reasons. The first is that robustness causes the existence of genotype networks--large connected sets of genotypes with the same phenotype. I discuss why genotype networks facilitate phenotypic variability. The second reason emerges from the evolutionary dynamics of evolving populations on genotype networks. I discuss how these dynamics can render highly robust phenotypes more variable, using examples from protein and RNA macromolecules. In addition, robustness can help avoid an important evolutionary conflict between the interests of individuals and populations-a conflict that can impede evolutionary adaptation.

Metabolic networks and their evolution

Since the last decade of the twentieth century, systems biology has gained the ability to study the structure and function of genome-scale metabolic networks. These are systems of hundreds to thousands of chemical reactions that sustain life. Most of these reactions are catalyzed by enzymes which are encoded by genes. A metabolic network extracts chemical elements and energy from the environment, and converts them into forms that the organism can use. The function of a whole metabolic network constrains evolutionary changes in its parts. I will discuss here three classes of such changes, and how they are constrained by the function of the whole. These are the accumulation of amino acid changes in enzyme-coding genes, duplication of enzyme-coding genes, and changes in the regulation of enzymes. Conversely, evolutionary change in network parts can alter the function of the whole network. I will discuss here two such changes, namely the elimination of reactions from a metabolic network through loss of function mutations in enzyme-coding genes, and the addition of metabolic reactions, for example through mechanisms such as horizontal gene transfer. Reaction addition also provides a window into the evolution of metabolic innovations, the ability of a metabolism to sustain life on new sources of energy and of chemical elements.

The molecular origins of evolutionary innovations

The history of life is a history of evolutionary innovations, qualitatively new phenotypic traits that endow their bearers with new, often game-changing abilities. We know many individual examples of innovations and their natural history, but we know little about the fundamental principles of phenotypic variability that permit new phenotypes to arise. Most phenotypic innovations result from changes in three classes of systems: metabolic networks, regulatory circuits, and macromolecules. I here highlight two important features that these classes of systems share. The first is the ubiquity of vast genotype networks - connected sets of genotypes with the same phenotype. The second is the great phenotypic diversity of small neighborhoods around different genotypes in genotype space. I here explain that both features are essential for the phenotypic variability that can bring forth qualitatively new phenotypes. Both features emerge from a common cause, the robustness of phenotypes to perturbations, whose origins are linked to life in changing environments.

The low cost of recombination in creating novel phenotypes: Recombination can create new phenotypes while disrupting well-adapted phenotypes much less than mutation

Recombination is often considered a disruptive force for well-adapted phenotypes, but recent evidence suggests that this cost of recombination can be small. A key benefit of recombination is that it can help create proteins and regulatory circuits with novel and useful phenotypes more efficiently than point mutation. Its effectiveness stems from the large-scale reorganization of genotypes that it causes, which can help explore far-flung regions in genotype space. Recent work on complex phenotypes in model gene regulatory circuits and proteins shows that the disruptive effects of recombination can be very mild compared to the effects of mutation. Recombination thus can have great benefits at a modest cost, but we do not understand the reasons well. A better understanding might shed light on the evolution of recombination and help improve evolutionary strategies in biochemical engineering.