Supplementary Material: An energy-speed-accuracy relation in complex networks for biological discrimination

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1 Suppementary Materia: An energy-speed-accuracy reation in compex networks for bioogica discrimination Feix Wong, 1,2 Arie Amir, 1 and Jeremy Gunawardena 2, 1 Schoo of Engineering and Appied Sciences, Harvard University, ambridge, MA 2138, USA 2 epartment of Systems Bioogy, Harvard Medica Schoo, Boston, MA 2115, USA We provide proofs here of the mathematica assertions made in the main text. I. EQUILIBRIUM ERROR FRATION FOR THE HOPFIEL MEHANISM The error fraction, ε, for the Hopfied mechanism is given in equation (4) of the main text, and is repeated here for convenience, ε = [ (k + m ) + m k ][(k + m )(W + ) + mk ] [ (k + m ) + m k ][(k + m )(W + ) + mk ]. (1) The background assumptions, as mentioned in the main text, are k = k, = and k /k < 1. If the mechanism is at thermodynamic equiibrium, then detaied baance must be satisfied. The equivaent cyce condition [1] appied to the two cyces in Fig. 1(a) of the main text yieds m m = k k = k k. (2) Note that W does not appear in equation (2) since, athough the mechanism itsef is at thermodynamic equiibrium, the system remains open, with substrate being converted to product. enote by ε eq the vaue of ε under the equiibrium constraint in equation (2). Using equation (2), define α, β so that α = = mk m k and β = = mk m k. Using the background assumptions, define the quantity ε, given by ε = α β = = k k, (3) to which a physica interpretation wi be given shorty. Substituting α and β into equation (1) and rewriting, we see that where ε eq = W.A + α W.B + β, k + m k + m A = (k + m ) + m k, B = (k + m ) + m k. orresponding author: jeremy@hms.harvard.edu If W =, then by equation (3), ε eq = ε, which shows that ε, as defined in equation (3), is the error fraction for the cosed system at thermodynamic equiibrium as defined in the main text. We now want to prove that ε eq increases from ε as W increases, for which it is sufficient to show that dε eq /dw >. For this, dε eq dw = Aβ Bα (W.B + β) 2, so that dε eq /dw > if, and ony if, A/B > α/β. We have ( A B = k + m ) ( (k + m ) + m k ) k + m (k + m ) + m k. (4) The foowing resut is straightforward. Lemma. onsider the rationa function r(x) = (a + x)/(b + x), where a, b >. If a/b > 1, then r(x) decreases stricty monotonicay from r() = a/b to 1. If a/b < 1, then r(x) increases stricty monotonicay from r() = a/b to 1. Appying the Lemma repeatedy to the terms in equation (4), and recaing the background assumptions, we see that k + m k + m > k k and (k + m ) + m k (k + m ) + m k > 1. Hence, A/B > k /k = α/β and so dε eq /dw >. It foows that ε eq increases stricty monotonicay from ε as W increases from. II. ERIVATION OF EQUATION (5) OF THE MAIN TEXT The non-equiibrium error fraction in equation (1) can be rewritten as ε(m ) = u(m )v(m ), where u(m ) = [ k + ( + k )m ] [ k + ( + k )m ], and v(m ) = [k ( + m + W ) + ( + W )m ] [k ( + m + W ) + ( + W )m ]. Using the background assumptions and the Lemma, we see that, as m increases, u(m ) decreases hyperboicay from u() = k /k = ε 1 to 1 whie v(m ) increases hyperboicay between + m + W + W v() = ε and v( ) =. + m + W + W

2 2 Hence, ε() = ( + m + W )/( + m + W ) and ε(m ) ( + W )/( + W ) as m. Furthermore, ε(m + m + W ) > v() = ε > ε 2, (5) + m + W as required for equation (5) in the main text. III. THE LIMITING ARGUMENT IN KINETI PROOFREAING The argument for kinetic proofreading given by Hopfied [2] is based on the non-dimensiona quantities, δ 1 = (m + k ) m k, δ 2 = m k, δ 3 = m, δ 4 = W, which are to be taken very sma. Accordingy, we consider the non-equiibrium error-fraction, ε, as defined in equation (1), in the imit as these four quantities. Since k > k, we have that δ 1 > (m + k ) m k >. (6) If we now divide above and beow by m k = m k in the expression ε = uv introduced above, we get ( ) δ ε = (m + k )/m k + 1 v. If we take δ 1 and formay treat v as a constant, we see from equation (6) that ε v as δ 1. However, the expression for δ 1 invoves m and this is aso a parameter in v. Hence, v is not constant during the imiting process, which has couped m to the vaues of the other parameters. If we ignore this couping, we can divide above and beow in v by k to get (1 + δ 2 )(W + ) + m v = (k /k + δ 2 )(W + ) + mk /k and if then et δ 2, we see that W + + m v ε W + + m If we now divide above and beow in this expression by, we get W/ + / + m/ W/ m/ ε 2 as δ 3 and δ 4. Hence, putting the sequence of imits together, we have, formay, im im im im ε = δ 4 δ 3 δ 2 δ ε2. 1 This seems to be the interpretation that has been given in the iterature to Hopfied s assertion that kinetic proofreading achieves the error fraction of ε 2.. The couping noted above specificay affects m, which has to satisfy two conditions. On the one hand m has to be arge, in order that u shoud be cose to u( ) = 1. That is the roe of δ 1. On the other hand m has to be sma, in order that v shoud be cose to v(). That is the roe of δ 2. The remaining imits for δ 3 and δ 4 are ony there to make sure that v() ε 2 ; compare equation (5). The consequence of the couping between the δ 1 and δ 2 imits, which arises through m, can be seen by rewriting δ 1, ( ) δ 1 = k k k Hence, in the imit as δ 1 and δ 2, k = k. In order to achieve the proofreading imit, it is necessary for rates other than m to change. Specificay, the on rates for the first discrimination, k = k, must become arge with respect to those for the second discrimination, =. IV. ERIVATION OF EQUATION (7) OF THE MAIN TEXT Suppose that R(x) and Q(x) are aowabe functions, as defined in the main text, with R(x)/x α c 1 and Q(x)/x β c 2, as x, where c 1, c 2 >. Since R 1 /x α (c 1 ) 1 >, it foows that R 1 is aowabe and deg(r 1 ) = deg(r). Since (RQ)/x α+β c 1 c 2 >, it foows that RQ is aowabe and deg(rq) = deg(r)+deg(q). Finay, suppose, without oss of generaity, that max(α, β) = α, so that α β. Then, R + Q x α = ( R x α ) + ( Q x β δ 2 ) x β α. (7) The imit of this, as x, is c 1 >, if α > β, or c 1 + c 2 >, if α = β. In either case, the imit is positive. Hence, R + Q is aowabe and deg(r + Q) = max(deg(r), deg(q)). This proves equation (7) of the main text. V. ERIVATION OF EQUATION (14) OF THE MAIN TEXT Suppose that S(x) is an aowabe function and that S/x α c >, where α = deg(s). Since n is a continuous function, n(s(x)) α n(x) n(c). ividing through by n(x), we see that n(s(x)) n(x) α. (8) If deg(s) = α >, then n(s) n(x), whie if deg(s) < then n(s) n(x 1 ), which proves equation (14) of the main text.

3 3 VI. ERIVATION OF EQUATION (15) OF THE MAIN TEXT 5 are constants independent of x. Since deg(ε) = 2, n(x) n(ε 1 )/2. Hence, Note that if R(x), Q(x), S(x), T (x) are functions, not necessariy aowabe, and if R Q and S T, so that R/Q c 1 > and S/T c 2 >, then (RS)/(QT ) c 1 c 2 >, so that RS QT. We wi use this without reference beow. Foowing the discussion in the main text, consider P (i j) = (R Q) n(r/q) where R and Q are aowabe functions with R/x α c 1 > and Q/x β c 2 >. There are three cases to consider. Suppose that α β. If α > β, then the same argument as in equation (7) shows that (R Q) x α. By equation (7) of the main text, deg(r/q) > 1, so that n(r/q) n(x). Hence, P (i j) x α n(x). If α < β, then (R Q) x β and n(r/q) n(x), so that P (i j) x β n(x). This proves case 1. Suppose that α = β but c 1 c 2. Then, (R Q)/x α c 1 c 2. Aso, ) x α R Q = ( R x α Q c 1 c 2 >. Hence, n(r/q) n(c 1 /c 2 ). Since (c 1 c 2 ) n(c 1 /c 2 ) >, it foows that P (i j) x α, which proves case 2. Suppose that α = β and c 1 = c 2. Then (R Q)/x α and R/Q 1, so that n(r/q). Hence, P (i j)/x α as x, so that P (i j) x α, which proves case 3. VII. ERIVATION OF EQUATION (2) OF THE MAIN TEXT For the Hopfied mechanism (Fig. 1(a) of the main text), we described in the main text how the asymptotic error rate of ε x 2 coud be achieved, by assuming that the abes are aowabe functions of x such that: deg( ) = deg( ) = 1, deg(k ) = deg( ) = 1 and deg(k ) = deg(k ) = deg(k ) = deg(m ) = deg(m) = deg( ) = deg(w ) =. Using equations (1) and (3) of the main text, we find that deg(p 1) = deg(p 2 ) = deg(p 3 ) =, deg(p 2 ) = 1, and deg(p 3 ) = 2. It is hepfu to introduce the notation R Q, for functions R(x), Q(x) which may not be aowabe, to signify that im x R/Q = 1. We can use this to cacuate the asymptotic behaviour of the terms P (i j) in the entropy production rate (equation (12) of the main text). For instance, ( P (1 3 ) = (( + W )p 3 p ( + W )p ) 3 1) n, p 1 The ony term in this expression which depends on x is for which deg( ) = 1. Since deg(n(r)) deg(r) n(x) (equation (8)), it foows that P (1 3 ) ( + W )p 3 n(x). Simiar cacuations yied P (2 3 ) 1, P (1 2 ) 2, P (1 3 ) 3, P (2 3 ) 4, and P (1 2 ) 5, where 1, 2, 3, 4, and so that P ( + W ) p 3 2 n(ε 1 ), im x P σ n(ε 1 ) = + W 2W. (9) This proves equation (2) of the main text. VIII. AITIONAL NUMERIAL ALULATIONS In Suppementary Figs. 1 and 2, we consider two discrimination mechanisms under the assumptions of equations (21) and (22) of the main text. Suppementary Fig. 1(a) shows a graph for McKeithan s T-ce receptor mechanism [3], whie Suppementary Fig. 2(a) shows a graph different from both this and the Hopfied exampe. We used previousy deveoped, freey-avaiabe software [4] to compute the Matrix-Tree formua (equation (1) of the main text) for each mechanism, from which we obtained symboic expressions for P, ε, and σ. The graphs in Suppementary Figs. 1(a) and 2(a) have 441 and 64 spanning trees rooted at each vertex, respectivey, underscoring the combinatoria compexity which arises away from equiibrium (main text, iscussion). (If a graph has reversibe edges, so that i j if, and ony if, j i, which is the case for a the graphs discussed here, there is a bijection between the sets of spanning trees rooted at any pair of distinct vertices.) Suppementary Figs. 1(b)-(c) and 2(b)-(c) show numerica pots undertaken in a simiar way to those for the Hopfied mechanism (main text, Fig. 2), as described in the main text. Simiar vertica and diagona bounds were found for the symmetric cases, whie simiar observations regarding the asymmetric cases as those made in the main text appy. IX. ASYMPTOTI RELATION FOR A NON-ISSOIATION-BASE MEHANISM We consider a discrimination mechanism having the graph shown in Suppementary Fig. 3(a). Its structure is identica to that of the Hopfied mechanism (Fig. 1(a) of the main text) but its abes differ to refect the energy andscape iustrated in Suppementary Fig. 3(b). If the abes are aowabe functions with deg( ) = 1 and deg( ) = deg( ) = deg( ) =, then, if the mechanism reaches thermodynamic equiibrium, it foows from equation (9) of the main text that its equiibrium error fraction satisfies ε eq x 1. (1) If it is further assumed that deg(m ) = deg(m ) = 1 and deg(m ) = deg(m ) = 1/2, whie a other abes have degree, then the mechanism is no onger at equiibrium. Using equations (1) and (3) of the main text, we find that

4 4 ρ 1 = [(m + k)( + W ) + km ][(m + k)( + W ) + km ] ρ 2 = [m ( + k ) + k ( + W )][(m + k)( + W ) + km ] ρ 3 = [m (k + ) + k ][(k + m )( + W ) + m k] ρ 2 = [(m + k)( + W ) + km ][m ( + k ) + k ( + W )] ρ 3 = [m (k + ) + k ][(k + m )( + W ) + m k]. It foows that deg(p 1) = deg(p 2 ) = deg(p 3 ) = deg(p 2 ) = (11) and that deg(p 3 ) = 3/2, (12) Using equations (7) and (14) of the main text aong with equations (11) and (12), we can cacuate the asymptotics of the terms in the entropy production rate P (equation (12) of the main text), assuming, as in the proof of the Theorem, that we are outside the measure-zero subset of parameter space arising from case 3 of equation (15) of the main text. We find that P (1 2 ) 1 P (2 3 ) x 1/2 n(x) P (1 3 ) 1 P (1 2 ) 1 P (2 3 ) x 1 n(x) P (1 3 ) x 1 n(x). It foows that P 1, so that the entropy production rate is asymptoticay constant or vanishes. Furthermore, it can be shown from equations (11) and (12) that deg(ε) = 3/2 and deg(σ) =. Hence, the error rate is asymptoticay better than at equiibrium, for which deg(ε eq ) = 1 (equation (1)), whie the speed remains asymptoticay constant. This refects a different asymptotic reation to that in equation (19) of the main text. [1] Gunawardena, J. A inear framework for time-scae separation in noninear biochemica systems, PLoS ONE 7, e36321 (212). [2] Hopfied, J. J. Kinetic proofreading: a new mechanism for reducing errors in biosynthetic processes requiring high specificity, Proc. Nat. Acad. Sci. USA 71, 4135 (1974). [3] McKeithan, T. W. Kinetic proofreading in T-ce receptor signa transduction, Proc. Nat. Acad. Sci. USA 92, 542 (1995). [4] Ahsendorf, T., Wong, F., Eis, R. & Gunawardena, J. A framework for modeing gene reguation which accommodates nonequiibrium mechanisms, BM Bio. 12, 12 (214). [5] Banerjee, K., Koomeisky, A. B. and Igoshin, O. A. Eucidating interpay of speed and accuracy in bioogica error correction, Proc. Nat. Acad. Sci. USA 114, 5183 (217).

5 5 (a) (b) 12 (c) n( n( FIG. 1. Numerics for the T-ce receptor mechanism. (a) Graph for an instance of McKeithan s T-ce receptor mechanism [3], with abe names omitted for carity. (b) Pot of P/σ against n(ε /ε) for approximatey 15 points, with the abes satisfying equations (21) and (22) of the main text and numericay samped as described in the main text. The vertica back dashed ine corresponds to the bound ε > ε4 for this mechanism (cacuation not shown) that is anaogous to equation (5) of the main text for the Hopfied mechanism. The diagona red dashed ine corresponds to equation (23) of the main text, as discussed further there. (c) Simiar pot to (b) but with interna discrimination between correct and incorrect substrates, as described in the text, with the ight bue points having a ower asymmetry range (A = 1) and the dark bue points having a higher range (A = 5). (a) P/σ (c) 4 P/σ (b) n(ε /ε) n(ε /ε) 4 FIG. 2. Numerics for another discrimination mechanism. (a) Graph for a discrimination mechanism that is different from both the Hopfied and McKeithan mechanisms, with abe names omitted for carity. (b) Points potted as in Suppementary Fig. 1(b). The vertica back dashed ine corresponds to the bound ε > ε2 for this mechanism (cacuation not shown) that is anaogous to equation (5) of the main text for the Hopfied mechanism. The diagona red dashed ine corresponds to equation (23) of the main text, as discussed further there. (c) Simiar pot to (b) but with interna discrimination between correct and incorrect substrates, as described in the text, with the ight bue points having a ower asymmetry range (A = 1) and the dark bue points having a higher range (A = 5).

6 6 (a) 3 m m +W 2 k k +W k m 1 2 k m 3 (c) 25 2 Hopfied mechanism.5.4 (b) energy 3 Δ 2 1 energy Δ 2 3 P/σ non-dissociation-based mechanism n(ε -1) FIG. 3. A non-dissociation-based mechanism. (a) Graph with the same structure as that for the Hopfied mechanism (Fig. 1(a) of the main text) but no discrimination between and takes pace through 1 2 X, whie interna discrimination takes pace through 2 X 3 X, as refected in the abe names. (b) Hypothetica energy andscape for the mechanism shown in (a), iustrating where energy may be expended to drive the steps with abes m and m. (c) Pot of P/σ against n(ε 1 ) for a numerica instance of the Hopfied mechanism (Fig. 1(a) of the main text, in bue) and a numerica instance of the non-dissociation-based mechanism in (a) (in red), as x is varied in the range x [, e 2 ]. The numerica abe vaues have been determined by taking k = = x and = = x 1 for the Hopfied mechanism and = m = m = x 1, m = x 1/2 and m = x 1/2 for the non-dissociation-based mechanism, with a other abes being 1. abe NAP ribosome (wid type) ribosome (hyperaccurate) ribosome (error-prone) k k k k m m 2.3 [ , 21.9]; 1 7 [6. 1 8, 16.6]; 1 7 [ , 17.5]; 1 7 m m [ ,.6];.272 [ ,.5];.299 [ , 2.1]; 1.85 W W TABLE I. Experimentay measured parameter vaues, in units of s 1, for the Hopfied mechanism in Fig. 1(a) of the main text, shown for discrimination during NA repication by the bacteriophage T7 NA poymerase (NAP) and discrimination during mrna transation by three E. coi ribosome variants, as annotated. The vaues were obtained from Tabes S1-S4 of [5]. The abes in the first coumn correspond to those in Fig. 1(a) of the main text, except that m, m and W now have subscripts and, for the correct and incorrect substrates, respectivey, to aow for interna discrimination, as expained in the main text. The vaues of m and were not known for the ribosome variants, so we chose m from m by randomy seecting n(m /m ) from the uniform distribution on [ 1, 1], which is simiar to the asymmetry ranges of the other parameters, and chose to satisfy the externa chemica potentia constraint used by [5], as expained in footnote [45] of the main text. The intervas given for m and indicate the range of samped vaues. Some sampes have ε < ε and these are not shown in Fig. 2(b) of the main text. The vaues foowing each interva give the averages of the potted vaues in Fig. 2(b) of the main text, as indicated there by asterisks, *.

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