Modelling bioactivities of combinations of whole extracts of edibles with a simplified theoretical framework reveals the statistical role of molecular diversity and system complexity in their mode of action and their nearly certain safety

Network pharmacology and polypharmacology are emerging as novel drug discovery paradigms. The many discovery, safety and regulatory issues they raise may become tractable with polypharmacological combinations of natural compounds found in whole extracts of edible and mixes thereof. The primary goal of this work is to get general insights underlying the innocuity and the emergence of beneficial and toxic activities of combinations of many compounds in general and of edibles in particular. A simplified model of compounds’ interactions with an organism and of their desired and undesired effects is constructed by considering the departure from equilibrium of interconnected biological features. This model allows to compute the scaling of the probability of significant effects relative to nutritional diversity, organism complexity and synergy resulting from mixing compounds and edibles. It allows also to characterize massive indirect perturbation mode of action drugs as a potential novel multi-compound-multi-target pharmaceutical class, coined Ediceuticals when based on edibles. Their mode of action may readily target differentially organisms’ system robustness as such based on differential complexity for discovering nearly certainly safe novel antimicrobials, antiviral and anti-cancer treatments. This very general model provides also a theoretical framework to several pharmaceutical and nutritional observations. In particular, it characterizes two classes of undesirable effects of drugs, and may question the interpretation of undesirable effects in healthy subjects. It also formalizes nutritional diversity as such as a novel statistical supra-chemical parameter that may contribute to guide nutritional health intervention. Finally, it is to be noted that a similar formalism may be further applicable to model whole ecosystems in general.

Feature sub-typing 126 Environmental exposure can be contemplated as exogeneous features which are imposed onto 127 the organism's endogenous features. For clarity of the discussion, features will be differentiated in 128 their notation relative to their nature: 129 -for environmental chemical exposure, e.g., nutrients and drugs, coined "N-type" features 130 with homeostatic (or normal/healthy) value of for nutrients that are necessary to maintain health 131 and with homeostatic value of 0 for any other chemical which is not required to maintain health, e.g., 132 "exotic" nutrients, drugs, pollutants, etc. We define the period for required dietary features. It will 133 range from typically 24 hours for "energetic" nutrients (e.g., sugar) to weeks for "structural" nutrients 134 (e.g., proteins) and "vitamins". Here we consider vitamins in a broad sense as being any chemical 135 substance necessary for health, thus seen as a broad class that may not be limited to vitamins in the 136 traditional sense. 137 -for disease causing agents, coined "D-type" features with homeostatic (or normal/healthy) 138 value of 0. The period for long term disease installation is coined and scales typically from weeks 139 to months. We differentiate between endogenous-disease features found only in patients, e.g., 140 mutations, injuries, accumulated toxins, etc., and exogeneous-disease features such as allergens and 141 biological agents, e.g., viruses, micro-organisms, parasites and, by convention in this work, tumors, 142 which can potentially all be cleared from the organism through therapeutics and the immune system. 143 -for organism's endogenous "E-type" features as defined earlier. In this work, we impose 144 that N-type and D-type features are not influenced by E-type features, i.e., they are imposed to the 145 organism by the environment (in a broad sense, including societal/medical influence, etc.). In doing 146 so, we leave out immunological and psychosomatic feedback loops between E-type and N-type and D-147 type features, e.g., the organism reacting to a shortage of nutrients, which may nevertheless be of 148 interest for further developments of the model, e.g., for exploring placebo/nocebo effects. Illness is 149 defined in this work as the occurrence of out-of-homeostasis state E-type features induced by excess or 150 shortage of N-type features and by D-type induced effects. 151 Some endogenous E-type features are compounds that are also found in the N-type 152 environmental chemical exposure, e.g., glucose is found in both blood and nutrients. Some exogenous 153 D-type features may also be found in different places. In those cases, we may consider again 154 compartmentation, e.g., the glucose in the digestive tract as an N-type feature will be differentiated 155 from the glucose level in other organism's compartments, which are E-type features. 156

Homeostasis perturbation
From there, the dependence of on { } { ≠ } when at least one is not at its homeostatic 161 value can be reduced to its first order approximation: 162 Introducing homeostasis perturbation due to nutritional perturbation transforms Equation 6 164 into: 165 In equation 7, , is a numerical value we coin the "potency" of the N-type feature and , is 167 a numerical value we coin the E-type "feature coupling". Both , and , are defined by the 168 organism's genetic/epigenetic makeup and microbiota makeup, and may thus potentially vary from 169 one individual to the other. Importantly, , and , can be as well positive as negative numerical 170 values, based on whether the effect of a compound will contribute to increase or decrease the feature 171 relative to its homeostatic value. 172 Because each E-type feature is itself modulated by the N-type features, we can rewrite 173 ( − ℎ 0 ) as: 174 Using equation 8 in equation 7 and reordering we obtain: 176 (9) 177 state homeostasis value ℎ for a healthy state taking into account a healthy nutritional regimen over 179 Here we suggest coining ⃛ , as the "potency" of the nutrient relative to the feature . 182 Secondly, when endogenous-disease-causing features are present, following the same route as 183 above, we obtain: 184 Because each E-type feature is itself modulated by the N-type and D-type features, we can 186 rewrite ( − ℎ ) as: 187 The latter further reduces to 191 where ⃛ , is a numerical value that we suggest coining the "virulence" of the D-type feature. The scaling of the effects of nutrients relative to their diversity. 210 We introduce the total quantity of nutrients as: 211 We can now rewrite as: 213 We introduce now the normalized activity of the nutrients on a given (E-type or G-type) 215 feature as: 216 Let's first put our attention to the scaling of the average of the magnitudes of ̃. In the 218 following, we contemplate  Noting that 〈 (̃) 2 〉 is the variance of ̃ (of mean 0), it becomes clear that: 226 We now consider the difference between two activities ̃ and ̃ distinguishable by their 228 differences in either ̅ , or . These activities can be written using random variables of mean 0 and 229 variance 2 : 230 ⃛ , ≡ ⃛ , (1 − 1 , ), ⃛ , ≡ ⃛ , (1 + 1 ) Following the previous approach, we readily find: 232 We now inspect ̅ , = ∑ , , taking into account that , can be considered as a random 234 variable of mean 0 and variance , . We designate by the number of features of the organism. The 235 latter is also related to the organism's complexity and therefore coined in this work as "complexity" . 236 With this convention we see that: 237 We can now focus on the absolute activity of the nutrients on a given (E-type or G-type) 239 feature as: 240 Following a route similar to the previous developments on ̃, we can readily conclude that: 242 From there we derive the following proportionality between the variance 2 and : 244 We can further introduce the effect of synergy as the nutritional diversity augments. In the 246 case of such synergy and for a given observed activity, the total amount is reduced to ⁄ , where is 247 then becomes 249 Now, at the expense of an acceptable loss of generality, we restrict the subsequent analysis to 251 the case where , and , fulfil the requirements for the applicability of the central limit theorem to 252 and ̃. Under these restrictions, ̃ can be approximated by a normal distribution. For any 253 organism and nutritional regimen, and are finite numbers, despite being unknown in practice. 254 We now consider given nutritional regimen, i.e., a mix of edibles. We assume that each edible 255 administrated individually at dose induces (on average) strong effects, i.e., cases where the 256 activity ̃ is greater than ̃ , where is a number set accordingly. When we administrate a 257 regimen at total dose composed of regimen as described above, each contributes individually at 258 dose / . Because individual nutritional regiments share a large proportion of chemical compounds, 259 the chemical diversity is increased solely by a factor < . As increased diversity is reflected by 260 replacing → in the model, the variance to consider now is ̃ 2

. 261
By assuming a normal distribution for ̃, we can evaluate numerically how ( ) varies 262 relatively to by first evaluating by numerically solving: 263 We then evaluate numerically the ratio between the average number of strong effects for the 265 mix of regimen of total dose relative and the average number of strong effects induced by each 266 individual regimen at dose : 267 Note that can be evaluated without explicitly specifying . 269

270
In this model, the action of any nutritional and pharmaceutical regimen is given by Equation 271 10. It is obtained by noticing through equation 7, 8 and 9 that the complex interrelationship between 272 features can be compounded formally into a single numerical factor ⃛ , , which can be as well positive 273 or negative. The latter, by its formal construction, is defined by the organism's genetic/epigenetic 274 makeup and microbiota makeup, and may thus potentially vary from one individual to the other. 275 The construction of Equation 10 shows also that the modulation by drugs or nutrients (N-type For real-world drugs and synthetic nutrients, but also for natural nutrients, is much larger 283 than .This is implicitly contemplated for natural nutrients because they are remaining features of 284 dead and processed organisms and thus necessarily comprise far fewer biological features than living 285 organisms. 286 To strictly maintain homeostasis, a "perfect regimen" is defined by = , which ensures 287 formally that ∑ ⃛ , = 0. In Equation 10, the latter sum is reflected into a single numerical value 288 ℎ . The detailed knowledge or definition of { } is thus not necessary. Evolution in a given 289 environment likely forged the numerical value of ℎ for each specie. In practice, pseudo-homeostatic 290 "optimal regimens" with ≠ are potentially achievable as long as ∑ ⃛ , ≈ 0 is verified for 291 every feature . N-type features imposed by the environment that are not necessary to maintain the 292 organism healthy, thus with = 0, need to verify also ∑ ⃛ , ≈ 0 in order to be without 293 noticeable effect, or in other words, in order to be tolerated by the organism. As ≫ , the latter 294 is possible only when a very particular relationship exists between {⃛ , } and { }. 295 An organism needs to access in sufficient amounts all the nutrients { } defining its 296 "homeostatic nutritional regimen" (HNR) during its dietary time . The latter can now be better 297 redefined as the time within which an organism needs to access its HNR for surviving over many 298 , i.e., being in good health. The contraposition is thus that if an organism does not access its HNR 299 over , undesired effects are likely to appear. 300 From Equation 14, disease symptoms relative to a given feature call to be defined as the 301 departure from the features homeostatic value: 302 The latter equation shows that in timeframes greater than , departures from the HNR in the 304 absence of disease conditions are symmetrical to disease conditions under strict HNR, i.e., nutritional 305 disequilibrium induced symptoms can be confounded with symptoms from disease causing features. It 306 is also clear that they can compensate each other, i.e., disease may be resolved by the suited non-HNR 307 for being all identical (in fact, in real cases they will likely be very different), it emerges that it is in 320 principle generally possible to assemble a set verifying simultaneously ∑ ⃛ , = 0 and 321 ∑ ⃛ , = ℎ . This shows that for exogeneous-diseases, in contrast to endogenous-diseases, there 322 is a possibility for nutritional regimen and compounds with a therapeutic activity (e.g., microbial 323 growth capacity inhibition) without deleterious effect in the absence of the disease. More realistically 324 on a numerical level, we may satisfyingly contemplate an optimal set of where ∑ ⃛ , ≈ 0 and 325 The scaling of effect of nutrients is mostly captured by equations 19 and 20 which 327 demonstrate that the most diverse the nutritional regimen of total amount (where each individual 328 compound is diluted as the diversity augments), the more likely that this regimen is without significant 329 effect on a randomly chosen feature. However, if we consider an additive multi-compound regimen 330 where the total amount augments with diversity (i.e., replacing by in the equations), such as in 331 poly-medication (whether with single chemical drugs or complex nutrients), the probability of 332 undesirable strong (side-)effects is predicted to increase, as expected. 333 Equation 22 indicates that the more diverse the nutritional regimen, i.e., the greater , the 334 most likely it is to maintain its effects (if any) in regards of small variations between organisms (e.g., 335 between individuals within a specie and between closely related species), and small variations in 336 nutrient composition (e.g., attributable to natural growth conditions of botanicals). and 26 and leads to a general scaling of the activity of chemical entities on features given by equation 339 27. It highlights a certain symmetry between the organisms' complexity and the diversity of a 340 nutritional regimen and the synergy that may arise as the diversity augments. Most importantly, these 341 effects can reinforce each other in decreasing the variance of the observed distribution of activities of 342 nutritional regimen. 343 The importance of this effect becomes apparent when this variance is further used in the 344 numerical evaluation of and given by equation 28 and 29. Remarkably, is only modestly 345 depending on both and the number of features (see Fig 1). The latter is varying only by a few 346 percent's when is varying over two decades, and is varying less than by a factor of 3 when is 347 varying over 12 decades. These results show that it is reasonable to consider a typical value for for 348 organisms of given complexity without the need to define precisely , i.e., without defining how 349 many features are actually significantly impacted when we consider an observable desired or 350 undesired effect, e.g., a side-effect. 351 In contrast to , varies dramatically with (Fig 2), a few percent of variation in being 353 enough to account for a decade variation in . This dependence on is increasing with , thus as 354 organisms increase in complexity. As a consequence, increasing diversity in a regimen will reduce 355 dramatically the number of significant effects in a complex organism relative to the reduction of 356 significant effects in an organism of lower complexity, and vice versa. As a rough numerical example 357 drawn from Fig 2, if we consider mixes of compounds, e.g., foods, that produce individually at a given 358 dose, e.g., 10 significant effects in both low and high complexity organisms, when the compound 359 diversity is increased by 20% in the mix, the low complexity organisms is expected to still experience 360 approximately 3 significant effects whereas for the high complexity organism the probability of 361 experiencing a single one is now only 1/10 th . 362 This scaling illustrates how, in this model, increasing the molecular diversity composing a 371 nutritional regimen dramatically decreases the probability of observing a significant activity on a 372 feature. The same applies when increased diversity leads to synergy for a desirable activity, and when 373 the complexity of the organisms augments due to the nutritional regimen, for instance through an 374 impact on microbiota diversity. 375

376
The analysis of this model has been restricted to a linear approximation of the deviation from 377 an average equilibrium value coined the homeostatic value. This precludes the description of the non-378 linear effects likely found in large deviation from homeostasis, notably the likely emergence of out-of-379 homeostasis equilibrium states. Nevertheless, a linear approximation has necessarily a validly up to 380 some point, especially when remembering that bimolecular association-dissociation between a ligand 381 and a target can be approximated by a linear equation within +/-10% accuracy for ligand 382 concentrations varying between 0 and 1.5Kd, the latter corresponding to 60% ligand binding. Also, 383 decomposing a single feature into two features, a first for the above-homeostasis values and a second 384 for the below-homeostasis values allows to formally use linear models to account for quadratic (or 385 higher degree) and asymmetrical effects induced on a dependent feature. This simplified and general model contributes to the understanding of why molecular 402 specificity of a single compound drug, though potentially a reality (e.g., with monoclonal antibodies), 403 is not to be considered, at least a priori, as an indicator for system level specificity [15]. Further 404 analysis may be considered to assess the possibility of single compounds with moderate but optimal 405 specificity to reduce the risk of side effects through the statistical averaging of the simultaneous 406 modulation of many features. This may then contribute to explain why several low specificity 407 pharmaceutical compounds, e.g., acetylsalicylic acid and acetaminophen [16], are surprisingly not 408 associated to excessive side effects relative to several putatively highly specific drugs, e.g., 409 MIPMADs compensate with effect ∑ , for the out-of-homeostasis term ∑ , of the disease 412 associated feature(s). As a consequence, when an endogenous-disease fighting drug is administrated to 413 a healthy individual, it results necessarily in inducing out-of-homeostasis term(s) to the disease 414 associated feature(s), which can be seen as side-effect symptoms (Equation 30). From there, a true 415 side effect appears better defined as the undesired modulation of one or several features different from 416 the endogenous-disease associated feature(s). We propose to differentiate side-effects by coining the 417 latter as "copathologic" and the former as "contrapathologic". Thus induced health benefits of high diversity relative to low diversity, contributes to call, besides many 435 other empirical observations described in the literature, to revisit and redesign nutritional and 436 microbiota studies in regards of global nutritional diversity intake [21]. From this analysis, nutritional diversity may not only affect directly the organisms itself, but be amplified by its action on the 438 diversity and complexity of the microbiota, which contributes indirectly and synergistically on the 439 molecular diversity provided in fine to the organism [22]. 440 Adaptation through evolution of an organism towards a particular nutritional environment 441 likely defines one of the possible organism's "perfect nutritional regimens", thus also the requirements 442 for an HNR, and its tolerance to environmental N-type features with = 0. This may actually be 443 observed for all organisms. Organisms living in a given stable nutritional environment likely share a 444 similar HNR and environmental tolerance. 445 Organisms adapted to highly diverse nutritional environments may not be able to access their 446 full HNR within one single meal duration or typical time , i.e., 24h for humans and most mammals. 447 Required food diversity may not be accessible within a single meal ration and/or a within the 448 geographical territory at reach during "meal duration". This may then impose that is much longer 449 than the characteristic time of such organism. At time scales smaller than such organisms are 450 permanently undergoing transient nutritional disequilibrium [23]. In order to achieve long-term 451 nutritional equilibrium and to avoid nutritional based disease building up over time, such organisms 452 need to continuously switch unequilibrated regimen at the scale of their typical time. As this 453 permanently out-of-equilibrium errancy around an equilibrium state bears some similarities with 454 walking with stilts, we coin it "stilts regimen". In the above equations ⃛ , are defined as resulting from the individual's genetic and 474 epigenetic/microbiota particularities. Individuals are thus not equal relative to food regimen. This is 475 trivially illustrated in the population particularities of the growing incidence of obesity or well 476 documented genetic related tolerance to alcohol, but this model shows that the concept is likely to be 477 generalized to all nutritional regimen. Nutrigenetics, similarly to pharmacogenetics, appear as a 478 necessity for matching nutritional regimen over a global homeostasis dietary period [26] 479 Returning to the potential equivalence between a single compound and a MIPMAD acting on 480 a given feature, this model makes non-HNR with therapeutic properties emerge as a therapeutic class 481 on its own. As with any other drug class, not every non-HNR is expected to be therapeutically 482 beneficial, neither toxic, and it activity may be significantly dependent on the organism's genetic 483 makeup. We propose to coin as "Ediceutical" an MIPMAD made of a complementation of HNR by an 484 excess of nutrients obtained from edibles and resulting in a therapeutic non-HNR, to be distinguished 485 from non-pharmaceutical "Nutraceuticals" and food supplementation to compensate nutritional 486 diversity deficiency. Another type of therapeutic non-HNR may be obtained by depleting an HNR, an 487 option which we will not explore further here. 488 an appropriate non-HNR or Ediceutical, nor how such curative activity may be systematically 490 identified, they definitely show that this can be as much a possibility as identifying a single chemical 491 compound with the desired curative properties. 492 Considering Ediceuticals as medical treatments raises the question of their potential toxicity 493 due to their complex and chemically uncharacterized nature. The scaling of revealed in Fig 2 shows  494 that for a mix of compounds the probability of significant effects diminishes dramatically as the 495 molecular diversity augments as long as the total dose (expressed in weight or moles) of compounds is 496 kept constant. By large empirical evidence, edibles are individually without significant risk of side-497 effects when ingested below a given individual "normal dose". The scaling of thus ensure that a 498 mix of edibles ingested each at no more than normal dose divided by will present even less risks of 499 side effects than each individual edible at its normal dose, regardless of whether or not this mix 500 enables a significant desirable effect, e.g., when this mix has been selected for a desirable therapeutic 501 effect. The latter likely holds also for extracts of edibles, especially whole extracts, e.g., the results of a 502 culinary process and/or digestive extracts, i.e., mostly a water extract of edibles possibly separated 503 from its solid remains. Additional fractionation, e.g., solvent extraction and essential oil production, 504 reduces molecular diversity and may concentrate toxins and break the system's toxicity equilibrium, 505 i.e., no longer verifying ∑ ⃛ , ≈ 0. Such potential increases in toxicities should be accounted for 506 when defining the normal dose of reference. 507 Besides the statistical effects related to potencies ⃛ , addressed by this model, the reduced 508 multi-molecular chemical reaction rate reduction due to the dilution of reactive compounds in the mix 509 will also contribute to reduce the risks of toxicities. As a numerical example, turning from a mix of 2 510 edibles to a mix of 20 edibles, at constant total dose, the chemical reaction rate between putative 511 reactive compounds that are uniquely found in each edible and compounds from the organism is 512 reduced by a factor 10, and between compounds found uniquely in different mixes by a factor 100. 513 is well experienced that almost every chemical library hit has a significant system level toxicity which 515 cannot always be compensated by medicinal chemistry. This means that the attrition rate of 516 Ediceuticals is expected to be very low relative to traditional novel chemical entities. 517 It is well known that botanical active principles can be found in very different amounts in 532 plants relative to geographical, meteorological and other environmental particularities, e.g., exposures 533 to pest attacks. Equation 22 shows that this is, intrinsically, much less likely an issue with MIPMAPs 534 in general and thus for Ediceuticals, in contrast to the empirical evidence found with traditional 535 botanical drugs made of highly fractionated botanical extracts. 536 The scaling of with the organism's complexity suggests that it may be possible to identify 537 MIPMAPs/Ediceuticals that have a significant effect on low complexity organisms and which are 538 without significant effect on a more complex organism solely because of the statistics of increased been shown that tumor cells have many impaired regulatory pathways [28]. This makes them less 541 complex than normal cells and a tumoral mass is obviously much less complex than an animal as a 542 whole. Differential complexity translates into differential robustness, and it appears in this model as a 543 novel class of therapeutic target [29]. 544 Finally, it is to be noted that the formalism developed here may be readily extended to whole 545 ecosystems in general by (re-)defining biological features accordingly. It may then contribute to a 546 better understanding on the role of biodiversity as such on individual biological features in a given 547 organism, e.g., how the loss of biodiversity induced by certain pesticides reflects on the health of 548 certain insects even when they are without noticeable direct effect on the latter [30]. 549

550
This work provides a very general model of the interplay between an organism and its 551 nutritional environment while it may be further adapted to model whole ecosystems in general. It 552 allows to characterize edibles as potential MIPMAPs and propose Ediceuticals as a potential novel 553 multi-compound-multi-target pharmaceutical class. Their mode of action may readily target 554 differentially organisms' system robustness as such based on differential complexities. It may be 555 leveraged for discovering nearly certainly safe novel antimicrobials and anti-cancer treatments. Such 556 an Ediceutical targeting Staphylococci spp. has already been exemplified in an animal model of 557 superficial skin infection [31]. This very general model provides also a general theoretical framework 558 to several pharmaceutical and nutritional observations. In particular, it characterizes two classes of 559 undesirable effects of drugs, and may question the interpretation of undesirable effects in healthy 560 subjects. It also formalizes nutritional diversity as such as a novel statistical supra-chemical parameter. 561 This may contribute to guide nutritional health intervention without the need to decipher all 562 underlining molecular details of food-heath interrelationship. 563