Maximizing output and recognizing autocatalysis in chemical reaction networks is NP-complete

Definitions

In the following paragraphs we formally introduce chemical reaction networks. We emphasize that our setup is the same as in the literature on flux analysis; we have opted, however, for a somewhat different notation that is closer to the conventions commonly used in graph theory as this makes the subsequent discussion more concise.

Reversible reactions are encoded by a pair of forward and backward reactions. The entries of the stoichiometric matrix are recovered as S_x,e= s_x,e+- s_x,e-.

In addition to the ordinary reactions like the one above, CRNs also contain pseudo-reactions E' representing influx and outflux of compounds of the form e_in(x)= ({x _in }, {x}) and e_out(x)= ({x}, {x _out }) where x _in and x _out refer to external reservoirs. These are additional vertices V' distinct from V. These pseudo-reactions feed the CRN and remove "waste products" and extract a desired output. In particular, the x _in , y _out ∈ V' do not take part in any other reaction.

We say that a species x is produced in a network if f(e_out(x)) > 0.

Note that this definition of f naturally generalized the definition of an (integer) flow on a directed graph with source x _in and target y _out , see e.g., [23]. In [26], a generalization of equ.(1), although restricted to hypergraphs with |e⁺| = 1, is considered, where the flows add up to a vertex-dependent demand term rather than to zero. In contrast to the usual setting of flow problems, we have a non-trivial restriction on the capacity only for the input edge(s), while the values of f are unrestricted for all other hyperedges.

Formulation of the problems

Outline

Formally, NP-completeness is defined for decision problems [27]. Optimization problems can be converted into decision problems by asking whether they admit a solution that is at least as good as some value. By abuse of language, it therefore makes sense to speak of an "NP-complete optimization problem" instead of using the phrase "the decision problem corresponding to our optimization problem is NP-complete".

The basic idea of proving that problem $\mathfrak{X}$ is NP-complete is to find a so-called reduction ρ from another problem $\mathfrak{B}$ that is already known to be NP-complete. The reduction ρ is an algorithm with polynomial runtime that converts any given instance of $\mathfrak{B}$ into an instance of $\mathfrak{X}$. An efficient (i.e., polynomial time) algorithm to solve (all instances of) $\mathfrak{X}$, therefore would also provide an efficient solution for every instance $P\in \mathfrak{B}$ by simply reducing P to $\rho \left(P\right)\in \mathfrak{X}$ then solving ρ(P). Hence we can conclude that $\mathfrak{X}$ is a hard problem when a known hard problem $\mathfrak{B}$ can be reduced to it.

In this section we devise a procedure that reduces every instance of the so-called 3-partition problem to a CRN with a single output pseudo-reaction in such a way that solving the output maximization problem for the CRN also solves the 3-partition problem. Thus optimizing output in CRNs is at least as hard as solving 3-partition. The same basic construction is then modified to show that the CRN can be built in such a way that all reactions are monomolecular or bi-molecular. We then employ the same construction to show that problem remains hard even if only a single source is provided. A simple modification finally establishes the hardness result for finding autocatalytic compounds.

3-Partition

The 3-partition problem (3PART) consists of deciding whether a given multiset of n = 3m integers s _i , i = 1, ..., 3m can be partitioned into triples that all have the same sum. This problem is one of the most famous strongly NP-complete problems, i.e., it stays NP-complete even when the numbers in the input instance are given in unary encoding [28], i.e., their values grows not faster than a polynomial in the problem size n. This remains true when the s _i are distinct [29]. If B denotes the desired sum of each subset then 3PART remains strongly NP-complete even if for every integer $B{/}_4<s_i<B{/}_2$ holds. The latter fact will be employed in our reduction proof in order to be able to show that an optimal mass flow through the network must have certain properties.

Basic Construction

Given an instance of 3PART we construct the associated CRN in a step-wise fashion. The first step is a lattice-like labeled graph, Figure 2(A), that consists of one input node for each s _i , m auxiliary nodes Z _j , each of which has an influx of $1/m_{}{\displaystyle {\sum }_i{s}_i}=s/m_{}$, an output sink node, 3m × m switch nodes, 3m waste nodes at the right and m waste nodes at the bottom. These switch nodes have two inputs; l from the left and u from above, and three outputs; r towards the right, d downwards, and o into the output channel. Each of the switch nodes can be in one of two distinct states: either it is

off The node transmits all its left input to right and all its input from above downwards, no flow is then diverted towards the output, i.e., r = l, d = u, o = 0; or

on The node consumes its entire input from the left (and thus transmits nothing to the right), at the same time uses up a corresponding amount of the input from above, and diverts the rest towards the output. Note that switch nodes are designed such that the flow downwards needs to be reduced by the same quantity as the flow to the right. As the flow to the right is completely consumed, i.e., the corresponding flow is reduced by l, it holds r = 0, d = u - l, o = l.

All flux along the output channel is collected in the output node, i.e., given a particular state of the switch nodes, the flux into the output node is the sum of the fluxes consumed from the left.

Lemma 1. An assignment of "on" and "off" to the 3m × m switch nodes is a solution of the original 3PART problem if and only if the total flow in the output node O equals the maximally possible value s = ∑ _i s _i .

Proof. Consider the CRN in Figure 2 with 3m × m switch nodes. Each column corresponds to one of the m desired subsets of the underlying instance of 3PART, each row corresponds to one of the 3m integer values s _i . Note that any assignment of "on" and "off" to switch nodes will split the overall horizontal as well as the overall vertical inflow into two parts: a part directed to waste material and an output part directed to node O. Let w _H (resp. w _V ) be the overall horizontally (resp. vertically) produced waste. For any assignment of "on" and "off" states to switch nodes s = f(e_out(O)) + w _H = f(e_out(O)) + w _V is invariant. Obviously, if w _H = w _V = 0, then the outflow f(e_out(O)) to node O is maximal. Furthermore note that at most one switch can be in "on" state in each row.

Consider an assignment of "on" and "off" to the switch nodes that corresponds to a solution of the original 3PART problem. Thus exactly 3m switch nodes are in mode "on" (three per column and one per row). As one switch node per row i is in mode "on", the outflux s _i of node Q _i flows to output node O and the waste produced horizontally in row i is 0. As this is true for all rows, w _H = w _V = 0 holds and the total flow in the output node O is s which is maximal.

Assume that the flow in the output node is the maximal possible value s, and therefore w _H = w _V = 0 holds. This implies that exactly one switch node per row needs to be in mode "on". As we can assume $s/{}_{4m}<s_i<s/{}_{2m}$ exactly 3 switch nodes per column need to be in state "on". The overall assignment is therefore a solution to the original 3PART problem.   □

Of course, the intermediate network in Figure 2(A) is not (yet) an proper CRN. To achieve this goal, we have to replace the switch nodes by hypergraphs that implement the high-level rule governing their behavior.

Implementing Switch-nodes

We see that the switch node (i, j) can be in the "on"-state only if it received at least s _i copies of the input from the left and a matching number of input molecules from above. A graphical description of this partial network is shown in Figure 2(B). Since the input from the left is limited to s _i copies of Q _i , either none or a single molecule of the intermediate X _ij is produced, depending on whether (i, j) is "on" or not. Clearly, for each i, only a single one of the switches (i, j) can be "on".

Note that equ.(5) already provides the necessary device to complete the proof. If we insist that the CRN may use at most bi-molecular reactions, we have to find a way to implement the reactions s _i Q _i → W _ij , X _ij → s _i O, and X _ij → s _i O by more restricted elementary reactions. This will the topic of the following section. According to equ.(5) each diamond node is replaced by 3(s _i +1) vertices, so that the entire network has $6m+2m+1+m{\sum }_{i=1}^{3m}3\left(s_i+1\right)=8m+3sm+3m^2+1$ nodes. Thus, all instances of 3PART for which s = s(m) is polynomially bounded in m can be reduced to a maximum output problem on an equivalent CRN. We explicitly use the fact that 3PART is strongly NP-complete: we need that m is polynomially bounded by the network size n to ensure that s, and thus the reduction to 3PART, remains polynomial. We know the maximal outflux of the CRN and can therefore use a simple guess-and-check argument to show that MAX-CRN-Output is in NP. Our discussion thus establishes

Theorem 1. MAX-CRN-Output is strongly NP-complete when the number of inputs into the CRN and number of educts in a chemical reaction is unrestricted.

We remark the our CRNs need to have at least two output nodes, one for the desired product and one to collect all waste products.

Restriction to Bi-molecular Reactions

In this section we show that the problem does not become easier when the CRN has only a single input and all reactions are bi-molecular. To this end we further refine the reactions s _i Q _i → W _ij , X _ij → s _i O, and X _ij → s _i O. We will make use of two specialized types of edges that can be implemented by bi-molecular reactions.

and the corresponding flow is chosen to be f ⁱ ({X _i , X_i+1}) := 2^u-i. Symmetrically, expansion-edges of type (1 → 2 ^u ) can be implemented by u subsequent reactions that split molecules repeatedly into two equal molecules. These (2 ^u → 1)-merge-edges (resp. (1 → 2 ^u )-expansion-edges) will in the following be used to implement the generalized merge- and expansion-edges.

Let b_m-1b_m-2... b₀ be the binary representation of k > 0 with m = ⌊log k⌋ + 1, and let B = {i₁, i₂, ..., i _r } be the indices of all non-zero bits, i.e., i ∈ B with b _i = 1. The underlying idea for the merging of k molecules of type I into 1 molecule of type O is to split the outflow k of I into r individual flows, i.e., $k={\sum }_{j=1}^r{2}^{i_j-1}$. We remark that this representation is unique. These flows of quantity $2^{i_j-1}$, j = 1, ..., r are then individually reduced to flows of size 1. The resulting r flows of quantity 1 are then all merged to a flow of one molecule of quantity 1. The implementation of generalized merge-edges is depicted in Figure 3(A). Expansion-edges that expand the flow of one molecule of quantity 1 to a flow of one molecule of quantity k can be implemented analogously. First, a flow of quantity 1 of one molecule is changed into r flows of quantity 1, then these r flows are expanded to r flows of quantity $2^{i_j-1}$, j = 1, ..., r, and then these flows are iteratively summed up. The details are depicted in Figure 3(B). Clearly, merge and expansion edges can be employed for the refinement of reactions s _i Q _i → W _ij , X _ij → s _i O, and X _ij → s _i O in equ.(5). The number of additional edges and nodes to implement a (k → 1) merge-edge is O(log² k), as there are O(log k) flows after the split into individual flows, and each individual flow employs O(log k) edges for the (k → 1) merge (with k being a power of 2). Symmetrically a (1 → k) expansion-edge uses O(log² k) bi-molecular edges and additional compounds. Based on this polynomial extension and as all merge and expansion reactions are bi-molecular, we have the following

Corollary 1. MAX-CRN(2)-Output is strongly NP-complete.

Restriction to a single input

To show that MAX-CRN-Output is NP-complete even if we have a single input only, we require an additional edge type that is implemented by connecting a (k → 1)-merge-edge and a (1 → k)-expansion edge in series. Such an edge ensures that exactly k (or exactly a multiplicity of k) input molecules react to the same number of output molecules. We will refer to these edges as (k)-force-flow-edges. Note, that such edges do not change the quantity of a flow. The number of additional edges and nodes required to implement a (k)-force-flow edge is O(log² k).

So far we assumed input nodes Q _i with corresponding influx s _i , i = 1, ..., 3m, plus the m additional input nodes Z₁, ..., Z _m with influx $s=1/m_{}{\displaystyle {\sum }_i{s}_i}$ each. In the following we will describe how to extend the construction of the CRN based on an instance of the 3PART problem (cmp. Figure 2) such that there is only a single input node. Note that all s _i , m, and the influx to nodes Z _i are defined by the given 3PART instance.

Influx to nodes Q _i

In the extended CRN the nodes Q _i will be internal nodes with influx s _i . In order to achieve this we will add a single input node Q with influx s', where s' is the integer representation of the concatenation of the r-bit binary representation of all s _i , i.e.,

$s^{\prime }=\sum _{i=1}^{3m}{s}_i\times 2^{r\left(i-1\right)},\mathsf{\text{with }}r=\max \left\{⌊\log s_i⌋\right\}+1$

(7)

Attached to node Q will be a subnetwork that splits the flux s' into the fluxes s₁, ..., s_3mby iteratively using the last r bits of the remaining flux as influx to a node Q _i , and then divide the remaining flux by 2 ^r . The hypergraph structure to implement this with bi-molecular reactions only is depicted in Figure 4. All dashed lines with red rectangles indicate force-flow-edges (the number in the rectangle indicates the enforced flow), all red edges with open arrowheads indicate merge- or expansion-edges. To enforce that exactly (and not a multiplicity) of s _i molecules flow towards node Q _i , the flow downwards needs to be maximized. This is done by introducing an additional outflux node: the flux of quantity s_3m≥ 1 towards O' is multiplied by a factor c, such that the additional overall non-waste outflux to O' dominates any other non-waste outflux. This can be ensured by choosing the factor c as the maximal possible influx to Q, i.e., c = 2^r×3m- 1 (the binary representation of c has r × 3m bit all set to 1). The number of additional edges and nodes is polynomially bound and the overall outflux of the extended network is then s_3m× c + ∑ _i s _i . As all outflux can be easily merged in a binary fashion as applied in the definition of expansion-edges, the resulting CRN has only a single input node and a single non-waste output node.

Autocatalysis

The construction of our reduction highlights the difficult part in determining autocatalysts. This is not so much finding the autocatalytic cycle itself but to ensure that the building blocks are provided from the "food source" through an in principle arbitrarily complicated sub-network.