Predicting anti-cancer activity in flavonoids: a graph theoretic approach

Abstract

In drug design, there are two major causes of drug failure in the clinic. First, the drug has to work, and second, the drug should be safe. Identifying compounds that work for certain ailments require enormous experimental time and, in general, is cost intensive. In this paper, we are concerned with melanoma, a special type of cancer that affects the skin. In particular, we seek to provide a mathematical model that can predict the ability of flavonoids, a vast and natural class of compounds that are found in plants, in reversing or alleviating melanoma. The basis for our model is the conception of a new graph parameter called, for lack of better terminology, graph activity, which captures melanoma cancer healing properties of the flavonoids. With a superior coefficient of determination, $R^2=1$, the new model faithfully reproduces anti-cancer activities of some known data-sets. We demonstrate that the model can be used to rank the healing abilities of flavonoids which could be a powerful tool in the screening, and identification, of compounds for drug candidates.

Introduction

In humans, a natural pigment, melanin, does not only provide a major defense mechanism against the ultraviolet light of the sun¹, but also determines the colour of the skin, hair and eyes². The production of melanin is catalyzed by the enzyme, tyrosinase. The overproduction of melanin leads to undesirable skin conditions, such as hyperpigmentation and melanoma skin cancer, that have an effect on the quality of life. The cancer, melanoma, has increasingly become a common complaint among patients consulting with dermatologists³.

Several agents, such as hydroquinone, corticosteroids, kojic acid and arbutin^3,4, that reduce the production of melanin by inhibiting the activity of the catalyst, tyrosinase, are well known. Unlike natural tyrosinase inhibitors, which are generally considered to be cheap and free of harmful side effects, the existing traditional anti-tyrosinase agents suffer a legion of limitations such as high levels of toxicity, low stability, poor skin-penetration, and insufficient activity (see, for instance³). The effectiveness of a therapeutic agent is generally measured by the inhibition concentration at $50\%$, $I{C}_{50}$, i.e., the quantity of the inhibitory agent required to inhibit the biological process, such as tyrosinase activity, by $50\%$⁵. The quantity $I{C}_{50},$ is synonymous to the half-saturation constant in ecology. The half-saturation constant is defined as the resource availability at which half of the maximum intake is reached and it determines the outcome of models and may contribute to explain behavioural traits, life-strategies and species occurrence⁶.

A well researched class of natural agents, flavonoids, commonly found in plants, have been reported to have anti-tyrosinase activity^1,2,5,7. Experiments to determine anti-tyrosinase activity of possible candidates for drug agents come with their own burdens. Among the burdens are high costs, laboratory experiment time, human efforts and the problems associated with animal sacrifice. Of particular concern however, as Hughes et al.⁸ puts it, drugs fail in the clinic for two main reasons; the first is that they do not work and the second is that they are not safe. Consequently, the development of mathematical models that can predict the effectiveness of compounds in inhibiting certain biological processes become handy.

In this paper, we will exploit graph theory to develop a model that predicts the anti-tyrosinase activity, i.e., $I{C}_{50}$ values, of flavonoids. Our model can be used, not only to rationalise existing data, but also to predict new or unknown anti-tyrosinase activity in flavonoids.

Graphs

A graph $G=(V,E)$ is a mathematical object which consists of a finite set V of elements called vertices, together with a set E, of 2-element subsets of V, called the edges of G. As early as 1875, Cayley (see for instance⁹), in his quest to enumerate chemical molecules called alkanes, he made an observation that molecules can be modelled by graphs where atoms are represented by vertices and two vertices are joined by an edge if the corresponding atoms are linked by bonds. This graph model became widely known as the molecular graph. The degree, deg v, of a vertex v of G is the number of edges incident with it. We say that v is an end vertex if its degree is 1. We will sometimes refer to the set of all end vertices as external vertices and call the set of all vertices of degree greater than 1 internal vertices. The irregularity index, t(v), of a vertex (introduced in¹⁰ and applied to studies in chemistry in¹¹) is defined as the number of neighbours of v with distinct degrees. The distance $d_G(u,v)$ between vertices u and v in G is defined as the length of a shortest path joining u and v in G. The eccentricity $\text{ec}(v)$ of a vertex v of G is the distance between v and a vertex furthest away from v in G.

The graph parameters

Consider a connected graph $G=(V,E)$ of order n, i.e., with n vertices. Over a hundred of graph parameters that are used to find relationships between the structure of a molecule and its physical properties in order to predict the physicochemical, biomedical, environmental and toxicological properties of a compound directly from its molecular structure are legion in the literature^12–17. It turns out that, to date, no relationships on the known parameters and anti-tyrosinase activity have been reported on. We create here a parameter that is broken down into two individual indices; one index specifically for external vertices, while the other one looks at internal vertices. We will then show that this new parameter can predict anti-tyrosinase activity.

Let S be the external vertices of G, i.e., the set of all end vertices in G. Let Q be the set of internal vertices of G. For a vertex v of G, the score, s(v), of v is the quantity

$$\begin{array}{c}\hfill s(v):=\left\{\begin{array}{cc}{\displaystyle {\sum }_{u\in S}d(u,v)}\hfill & \text{if}v\in S,\hfill \\ \hfill \\ \frac{t(v)}{\text{deg}v}·\text{ec}(v)\hfill & \text{if}v\in Q.\hfill \end{array}\right)\end{array}$$

We define two invariants, D(G) and $\zeta (G)$, of G as follows:

$$\begin{array}{c}\hfill D(G)=\frac{{\sum }_{v\in S}s(v)}{n^3},\end{array}$$

and

$$\begin{array}{c}\hfill \zeta (G)=\frac{{\sum }_{v\in Q}s(v)}{n^2}.\end{array}$$

For lack of better terminology, we will call D(G) the external activity of G, and $\zeta (G)$, the internal activity of G, respectively. For instance, for the molecular graph T of the known standard tyrosinase inhibitor depicted in Fig. 1, Taxfolin, we have

$$\begin{array}{c}\hfill D(T)=\frac{196}{{22}^3}=0.0184072\text{and}\zeta (T)=\frac{85.8333}{{22}^2}=0.1773415.\end{array}$$ 1

Figure 1.

Tyrosinase inhibitor, Taxifolin (5,7,3${}^{\prime }$,4${}^{\prime }$-flavan-on-ol), also known as dihydroquercetin.

The model

Consider a flavonoid F whose molecular graph is G. We combine the internal and external activities of G to predict the anti-tyrosinase activity, i.e., $I{C}_{50}$ value, of F. Table 1 presents flavonoids which were isolated from a plant in¹. We will capitalise on this data-set to determine the values of the constants for our model, (2). We compute D(G) and $\zeta (G)$ for each of the flavonoid in Table 1. The graphs are shown in Fig. 2.

Table 1.

Data for nine flavonoids.

Flavonoid	D(G)	$\zeta (G)$	${\mathit{IC}}_{50}$ ($\mu$M)
Dihydroquercetin-4${}^{\prime }$-methylether	0.016848853	0.187145558	115
Dihydroquercetin-7,4${}^{\prime }$-dimethylether	0.015552662	0.1953125	162
5,7,3${}^{\prime }$,5${}^{\prime }$-Tetrahydroxyflavanone	0.014685239	0.155706727	423
Blumeatin	0.013429752	0.164600551	624
Quercetin	0.018407213	0.177341598	96
Rhamnetin	0.016931043	0.187775677	107
Tamarixetin	0.016931043	0.193761815	144
Luteolin	0.015117158	0.197656841	258
Luteolin-7-methyl ether	0.013899324	0.202479339	350

Figure 2.

Molecular graphs of isolated flavonoids, Source¹.

In Table 1, we present D(G) and $\zeta (G)$ of the flavonoids, together with their anti-tyrosine activities, which were determined in¹.

Given that the activity is driven by two invariants, D(G) and $\zeta (G),$ we propose a multivariable function to model the effects of the two invariants on the $I{C}_{50}$ values. We propose a model of the form

$$\begin{array}{cc}\hfill i{c}_{50}(G)=& f([D(G)],[\zeta (G)])\hfill \\ \hfill =& {\alpha }_1+{\alpha }_2[D(G)]+{\alpha }_3[\zeta (G)]+{\alpha }_4{[D(G)]}^2+{\alpha }_5[D(G)][\zeta (G)]\hfill \\ \hfill & +{\alpha }_6{[\zeta (G)]}^2+{\alpha }_7{[D(G)]}^2[\zeta (G)]+{\alpha }_8[D(G)]{[\zeta (G)]}^2+{\alpha }_9{[\zeta (G)]}^3.\hfill \end{array}$$ 2

Fitting (2) to data given in Table 1, we obtain the values of the constants as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\alpha }_1& =4.571\times {10}^5,{\alpha }_2=-9.612\times {10}^7,{\alpha }_3=1.027\times {10}^6,{\alpha }_4=1.197\times {10}^9,\hfill \\ \hfill {\alpha }_5& =8.704\times {10}^8,{\alpha }_6=-4.565\times {10}^7,{\alpha }_7=-7.457\times {10}^9,{\alpha }_8=-1.715\times {10}^9,\hfill \\ \hfill {\alpha }_9& =1.363\times {10}^8.\hfill \end{array}\end{array}$$

The given parameter values ${\alpha }_i,i=1,\dots ,9,$ produce a perfect fit of the model to the data. It is important to note that we resorted to the multivariable polynomial since it gives the best goodness of fit value measured by $R^2$ or R-square. The strength of the relationship between a model and the dependent variables is measured by $R^2\in [0,1]$ or on a scale of $0-100\%.$ We also note that the Curve Fitting Toolbox that we used in fitting our data to the model, gives a number of goodness of fit statistics for parametric models and in particular, the sum of squares due to error (SSE) and R-square. The summed square of residuals (SSE) is a measure of the total deviation of the response values from the fit to the experimental data. Our fit gives: SSE $=2.609\times {10}^{-19}$ and $R^2=1.$

A consideration of the contour map of Fig. 3 gives the results in Fig. 4. All the data points lie in the same contour colour indicating a perfect fit of the polynomial to the data.

Figure 3.

Show the multivariable polynomial fit to the data.

Figure 4.

Shows the contour plot of D(G) and $\zeta (G)$.

A plot of the residuals is shown in Fig. 5. The residuals are found by looking at the difference between the data and the fit. The smaller the difference, the better the fit. The graph shows that the data points lie (almost) on the $D(G)\zeta (G)-$trace, indicating very low values of the residuals.

Figure 5.

Show the plot of the residuals, where residuals=data-fit.

Discussion and applications

Our model has been developed, not only to rationalise existing data, but also to predict new or unknown anti-tyrosinase activity in flavonoids. However, for the sake of brevity and with such a very high goodness of fit, we use the newly developed model to rank the order of inhibition for 26 flavonoids. Additional flavonoids can, nevertheless, be added to the ranking list by using the model. The results are presented in Table 2.

Table 2.

26 Flavonoids - rank order of inhibition.

NO	COMPOUND	D(G)	$\zeta (G)$	PREDICTED ${\mathit{IC}}_{50}(\mu \text{M})$
1	Kojic acid	0.028	0.238333	$-$ 112089.011
2	Chrysin	0.004082228	0.202216066	$-$ 40655.55419
3	Shikonin	0.028074722	0.193499615	$-$ 33061.25049
4	Baicalein	0.00625	0.19833325	$-$ 26435.10902
5	Galangin	0.00725	0.18	$-$ 14014.27835
6	Dihydromyricetin	0.022355552	0.149968431	$-$ 9665.559405
7	Naphthazarin	0.020408163	0.193877551	$-$ 3748.192859
8	Xanthoxylin	0.036443149	0.147959184	$-$ 2455.072672
9	Tropolone	0.008230453	0.222222222	$-$ 1193.329897
10	Morin	0.017843727	0.170110186	35.71991087
11	Quercetin	0.018407213	0.177341598	93.33756038
12	Quercetin-3-rutinoside	0.018407213	0.177341591	93.33764064
13	Taxfolin	0.018407213	0.177341529	93.33836295
14	Rhamnetin	0.016931043	0.187775677	105.5945071
15	Dihydroquercetin-4${}^{\prime }$-methylether	0.016848853	0.187145558	113.0227843
16	Tamarixetin	0.016931043	0.193761815	147.3433049
17	Dihydroquercetin-7,4${}^{\prime }$-dimethylether	0.015552662	0.1953125	162.703958
18	Luteolin	0.015117158	0.197656841	259.6199685
19	Luteolin-7-methyl ether	0.013899324	0.202479339	352.5847798
20	5,7,3${}^{\prime }$,5${}^{\prime }$-Tetrahydroxyflavanone	0.014685239	0.155706727	427.3532608
21	Blumeatin	0.013429752	0.164600551	617.4170072
22	Apigenin	0.0105	0.219166675	4333.366225
23	Fisetin	0.01425332	0.212018141	4757.002259
24	Rosmarinic acid	0.020197997	0.250493047	6636.14232
25	3,7,4${}^{\prime }$-Trihydroxyflavone	0.01025	0.235	27875.46695
26	Isoeugenol	0.020833333	0.275462986	33581.06508

Quite exciting is the fact that whilst the second compound, among the ranked flavonoids, chrysin, is commonly being used for bodybuilding, treating anxiety, inflammation, gout, erectile dysfunction, to mention but a few, it is appearing on the list as an excellent anti-tyrosinase agent which is only 3-fold weaker, and 34-fold stronger than the standard tyrosinase inhibitors, kojic acid and tropolone⁴, respectively. Whilst these results on chrysin confirm Wu et al.¹⁸’s findings that chrysin has potential use in skin photoprotection, Xie et al.¹⁹ found, in experimental work, that ‘chrysin had no effects on the (tyrosinase) enzyme’.

It is interesting to note that, through expanding the ranking list, our model can be used as a tool for identifying flavonoids that can be candidates for anti skin-cancer drug agents. This is critical in drug development as new drug agents get identified. Any identified agents can be tested using the model to determine their likelihood of use in fighting skin-cancer in the absence of laboratory tests.

Finally, we note here that it will be a worthwhile exercise to investigate the mathematical properties of the external, and internal activities of a graph, especially lower and upper bounds, and characterising the extremal graphs. Whilst external vertices play a significant role in influencing the anti-tyrosinase activity, the most dominant factor is the contribution made by the internal vertices. Each internal vertex v’s contribution is based on the distance between v and a vertex furthest away from v. This distance is then amplified by the irregularity of the neighbours of v.

The work presented in this paper is limited by the data set that is small. Predictive models depend a lot on the size of the data set. This model only predicts for flavonoids yet there may be other compounds that could be used as anti-cancer drugs. In reality, however, the class of flavonoids is much bigger than presented here. Despite these short comings, the model presented here forms a very good basis of predictive modelling in drug design. The model managed to confirm the effectiveness of some of the most used anticancer drugs such as tropolone and kojic acid.

Author contributions

S.M. and F.N. wrote the main manuscript. S.M. wrote the graph-theoretic section and F.N. wrote the modelling section. All authors reviewed the manuscript.

Data availability

All data generated or analysed during this study are included in this published article.

Competing interests

The authors declare no competing interests.