Exhibit 99.2

Lock-Up Agreement

May 21, 2018

COWEN AND COMPANY, LLC

PIPER JAFFRAY & CO.

As Representatives of the several Underwriters

c/o Cowen and Company, LLC

599 Lexington Avenue

New York, New York 10022

c/o Piper Jaffray & Co.

800 Nicollet Mall, Suite 1000

Minneapolis, Minnesota 55402

Re: Sutro Biopharma, Inc. – Registration Statement on Form S-1 for Shares of Common Stock

Dear Sirs or Madams:

This Agreement is being delivered to you in connection with the proposed Underwriting Agreement (the “Underwriting Agreement”) between Sutro Biopharma, Inc., a Delaware corporation (the “Company”) and Cowen and Company, LLC (“Cowen”) and Piper Jaffray & Co. (“Piper”), as representatives (the “Representatives”) of a group of underwriters (collectively, the “Underwriters”), to be named therein, and the other parties thereto (if any), relating to the proposed public offering of shares of the common stock (the “Offering”), par value $0.001 per share (the “Common Stock”), of the Company.

In order to induce you and the other Underwriters to enter into the Underwriting Agreement, and in light of the benefits that the offering of the Common Stock will confer upon the undersigned in its capacity as a securityholder and/or an officer, director or employee of the Company, and for good and valuable consideration, the receipt and sufficiency of which are hereby acknowledged, the undersigned agrees with each Underwriter that, during the period beginning on the date hereof through and including the date that is the 180th day after the date of the Underwriting Agreement (the “Lock-Up Period”), the undersigned will not, without the prior written consent of Cowen and Piper, directly or indirectly, (i) offer, sell, assign, transfer, pledge, contract to sell, or otherwise dispose of, or announce the intention to otherwise dispose of, any shares of Common Stock (including, without limitation, Common Stock which may be deemed to be beneficially owned by the undersigned in accordance with the rules and regulations promulgated under the Securities Act of 1933, as amended (such shares, the “Beneficially Owned Shares,” and such act, the “Securities Act”)) or securities convertible into or exercisable or exchangeable for Common Stock, (ii) enter into any swap, hedge or similar agreement or arrangement that transfers in whole or in part, the economic risk of ownership of the Beneficially Owned Shares or securities convertible into or exercisable or exchangeable for Common Stock, whether now owned or hereafter acquired by the undersigned or with respect to which the undersigned has or hereafter acquires the power of disposition, or (iii) engage in any short selling of the Common Stock or securities convertible into or exercisable or exchangeable for Common Stock.

If the undersigned is an officer or director of the Company, the undersigned further agrees that the foregoing provisions shall be equally applicable to any issuer-directed shares of Common Stock the undersigned may purchase in the Offering.

If the undersigned is an officer or director of the Company, (i) Cowen and Piper agree that, at least three business days before the effective date of any release or waiver of the foregoing restrictions in connection with a transfer of shares of Common Stock, Cowen and Piper will notify the Company of the impending release or waiver, and (ii) if required by FINRA Rule 5131 (or any successor provision thereto), the Company has agreed or will agree in the Underwriting Agreement to announce the impending release or waiver with respect to the undersigned by press release through a major news service at least two business days before the effective date of the release or waiver. Any release or waiver granted by Cowen and Piper hereunder to any such officer or director shall only be effective two business day after the publication date of such press release. The provisions of this paragraph will not apply if (a) the release or waiver is effected solely to permit a transfer not for consideration and (b) the transferee has agreed in writing to be bound by the same terms described in this Agreement to the extent and for the duration that such terms remain in effect at the time of the transfer.

The restrictions set forth in the second paragraph shall not apply to:

(1)     if the undersigned is a natural person, any transfers made by the undersigned (a) as a bona fide gift, or gifts, or for bona fide estate planning purposes, (b) to any member of the immediate family (as defined below) of the undersigned or to a trust the direct or indirect beneficiaries of which are exclusively the undersigned or members of the undersigned’s immediate family, (c) by will, testamentary document or intestate succession upon the death of the undersigned, or (d) as a bona fide gift to a charity or educational institution; provided, however, (A) in the case of any transfer described in clauses (1)(a) and (1)(b), no filing by any party (donor, donee, transferor or transferee) under Section 16 of the Securities Exchange Act of 1934, as amended (the “Exchange Act”) except a filing on a Form 5, but no sooner than February 10, 2019, or other public announcement shall be required or shall be made voluntarily in connection with such transfer or distribution, and (B) in the case of any transfer described in clauses (1)(c) and (1)(d), if the undersigned is required to file a report under Section 16 of the Exchange Act reporting any transfer pursuant to clauses (1)(c) or (1)(d), the undersigned shall include a statement in such report to the effect that such transfer relates to the circumstances described in clauses (1)(c) and (1)(d), as the case may be, and no other public announcement shall be required or shall be made voluntarily in connection with such transfer or distribution.

(2)    if the undersigned is a corporation, partnership, limited liability company or other business entity, any transfers to any stockholder, partner (which for additional clarity, includes limited partners) or member or managers of, or owner of a similar equity interest in, the undersigned, as the case may be, or to the estates of any such stockholders, partners, members, managers, or owners of similar equity interest in the undersigned, if, in any such case, such transfer is not for value;

(3)     if the undersigned is a corporation, partnership, limited liability company or other business entity, any transfer made by the undersigned (a) in connection with the sale or other bona fide transfer in a single transaction of all or substantially all of

the undersigned’s capital stock, partnership interests, membership interests or other similar equity interests, as the case may be, or all or substantially all of the undersigned’s assets, in any such case not undertaken for the purpose of avoiding the restrictions imposed by this Agreement or (b) to another corporation, partnership, limited liability company or other business entity so long as the transferee is a direct or indirect affiliate (as defined below) of the undersigned and such transfer is not for value;

(4)    if the undersigned is a trust, to a trust, trustee or beneficiary of the trust or to the estate of a trustor, trustee or beneficiary of such trust;

(5)    the transfer of Common Stock or any security convertible into or exercisable or exchangeable for Common Stock (or the economic consequences of ownership of Common Stock) that occurs pursuant to a settlement agreement related to the distribution of assets in connection with the dissolution of a marriage or civil union, by operation of law pursuant to a qualified domestic order in connection with a divorce settlement or pursuant to any other court order;

(6)    transactions relating to Common Stock or other securities convertible into or exercisable or exchangeable for Common Stock acquired in the Offering or in open market transactions after completion of the Offering; provided, however, that no filing under Section 16 of the Exchange Act or other public announcement shall be required or shall be voluntarily made in connection with subsequent sales of Common Stock or other securities acquired by the undersigned in such open market transactions;

(7)     to the Company pursuant to the undersigned’s employment agreement or agreements governed by the Company’s equity incentive plans described in the final prospectus relating to the Offering (the “Prospectus”) under which the Company has the option to repurchase such shares or a right of first refusal with respect to transfers of such shares upon termination of service of the undersigned;

(8)    the establishment of a trading plan pursuant to Rule 10b5-1 of the Exchange Act, provided, however, (i) that such plan does not provide for, or permit, the sale of any Common Stock during the Lock-up Period, (ii) the establishment of such plan is not required to be reported in any public report or filing with the SEC, or otherwise, and (iii) the undersigned does not otherwise voluntarily effect any public filing or report or any public announcement regarding the establishment of such trading plan;

(9)     any transfers made by the undersigned to the Company to satisfy tax withholding obligations pursuant to the Company’s equity incentive plans or arrangements disclosed in the Prospectus; provided, however, that no filing under Section 16 of the Exchange Act or other public filing, report or announcement reporting a reduction in beneficial ownership of shares of common stock shall be required or shall be voluntarily made during the 30 days after the date of the Prospectus, and after such 30th day, if the undersigned is required to file a report under Section 16 of the Exchange Act reporting a reduction in beneficial ownership of shares of common stock during the Lock-up Period, the undersigned shall clearly indicate in the footnotes thereto that the filing relates to the circumstances described in this clause (9);

(10)    the transfer of shares of Common Stock or any security convertible into or exercisable or exchangeable for Common Stock pursuant to a bona fide third-party tender offer, merger, consolidation or other similar transaction made to all holders of the Company’s securities involving a change of control of the Company that is approved by the board of directors of the Company, provided, however, that in the event that such tender offer, merger, consolidation or other such transaction is not completed, such securities held by the undersigned shall remain subject to the restrictions on transfer set forth in this Agreement;

(11)     the conversion or reclassification of the outstanding preferred shares or other securities of the Company into Common Stock in connection with the consummation of the Offering as described in the Prospectus, provided, however, that any such Common Stock received upon such conversion or reclassification shall be subject to the restrictions contained in this Agreement; and

(12)     transactions pursuant to the Underwriting Agreement;

provided, however, that (A) in the case of any transfer described in clause (1) through (5) above, it shall be a condition to the transfer that the transferee executes and delivers to Cowen and Piper, acting on behalf of the Underwriters, not later than one business day prior to such transfer, a written agreement, in substantially the form of this Agreement (it being understood that any references to “immediate family” in the agreement executed by such transferee shall expressly refer only to the immediate family of the undersigned and not to the immediate family of the transferee), (B) in the case of any transfer described in clauses (2) through (4), no filing by any party (donor, donee, transferor or transferee) under Section 16 of the Exchange Act or other public announcement shall be required or shall be made voluntarily in connection with such transfer or distribution during the Lock-Up Period, (C) in the case of any transfer described in clauses (5) or (7), any required filings made under Section 16 of the Exchange Act shall state that the transfer is by operation of law, court order, in connection with a divorce settlement, a repurchase by the Company or the exercise of the Company’s right of first refusal, as the case may be, and no other public announcement shall be required or voluntarily made, and (D) in the case of any transfer described in clauses (1) or (4) such transfer shall not involve a disposition for value. For purposes of this paragraph, “immediate family” shall mean a spouse or domestic partner, child, grandchild or other lineal descendant (including by adoption), father, mother, brother or sister of the undersigned; and “affiliate” shall have the meaning set forth in Rule 405 under the Securities Act. For the purposes of clause (10), “change of control” shall mean the transfer (whether by tender offer, merger, consolidation or other similar transaction), in one transaction or a series of related transactions, to a person or group of affiliated persons (other than an Underwriter pursuant to the Offering), of the Company’s voting securities if, after such transfer such person or group of affiliated persons, other than the Company or its subsidiaries, becomes the beneficial owner (as defined in Rules 13d-3 and 13d-5 of the Exchange Act) of 90% or more of the outstanding voting securities of the Company (or the surviving entity).

For avoidance of doubt, nothing in this Agreement prohibits the receipt by the undersigned of shares of Common Stock upon the exercise or settlement of any options, restricted stock units, warrants or other rights (which exercises may be effected on a cashless or net exercise basis to the extent permitted by the instruments representing such security to cover the exercise price or taxes due upon the exercise or vesting of such security), it being understood that any Common Stock issued upon such exercises or settlement will be subject to the restrictions of this Agreement.

In order to enable this covenant to be enforced, the undersigned hereby consents to the placing of legends or stop transfer instructions with the Company’s transfer agent with respect to any Common Stock or securities convertible into or exercisable or exchangeable for Common Stock except in compliance with the foregoing restrictions.

The undersigned further agrees that, without the prior written consent of Cowen and Piper on behalf of the Underwriters, it will not, during the Lock-Up Period (as the same may be extended as described above), make any demand or request for or exercise any right with respect to the registration under the Securities Act, of any shares of Common Stock or other Beneficially Owned Shares or any securities convertible into or exercisable or exchangeable for Common Stock or other Beneficially Owned Shares. For the avoidance of doubt, the undersigned hereby waives any and all notice requirements and rights with respect to the registration of any securities pursuant to any agreement, instrument, understanding or otherwise setting forth the terms of any security of the Company held by the undersigned, including any stockholders or registration rights agreement or similar agreement, to which the undersigned is a party or under which the undersigned is entitled to any right or benefit; provided, however, that such waiver shall apply only to the proposed Offering, and any other action taken by the Company in connection with the proposed Offering.

This Agreement and all authority herein conferred are irrevocable and shall survive the death or incapacity of the undersigned and shall be binding upon the heirs, personal representatives, successors and assigns of the undersigned.

The undersigned hereby represents and warrants that the undersigned has full power and authority to enter into this Agreement and that this Agreement has been duly authorized (if the undersigned is not a natural person), executed and delivered by the undersigned and is a valid and binding agreement of the undersigned.

This Agreement shall be governed by and construed in accordance with the internal laws of the State of New York applicable to agreements made and to be performed in such state.

If (i) the Company notifies Cowen and Piper in writing that it does not intend to proceed with the Offering, (ii) the Company files an application with the Securities and Exchange Commission to withdraw the registration statement related to the Offering, (iii) the Underwriting Agreement is not executed by December 31, 2018 (provided, however, that the Company may extend such date by up to three months with written notice to the undersigned prior thereto if the Company is still pursuing the Offering contemplated by the Underwriting Agreement), or (vi) the Underwriting Agreement (other than the provisions thereof which survive termination) shall terminate or be terminated for any reason prior to payment for and delivery of any Common Stock to be sold thereunder, then this Agreement shall immediately be terminated and the undersigned shall automatically be released from all of his or her obligations under this Agreement. The undersigned acknowledges and agrees that whether or not any public offering of Common Stock actually occurs depends on a number of factors, including market conditions.

The undersigned hereby consents to receipt of this Agreement in electronic form and understands and agrees that this Agreement may be signed electronically. In the event that any signature is delivered by facsimile transmission, electronic mail, or otherwise by electronic transmission evidencing an intent to sign this Agreement, such facsimile transmission, electronic mail or other electronic transmission shall create a valid and binding obligation of the undersigned with the same force and effect as if such signature were an original. Execution and delivery of this Agreement by facsimile transmission, electronic mail or other electronic transmission is legal, valid and binding for all purposes.

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